Dashboard Mechanisms for Online Marketplaces

TL;DR

This paper introduces a theoretical framework for designing dashboard-based mechanisms in online marketplaces, enabling long-lived agents to optimize bids with minimal calls to the allocation algorithm.

Contribution

It develops online winner-pays-bid and all-pay mechanisms that use dashboards to replicate truthful outcomes with strategic equivalence, reducing algorithm calls.

Findings

Dashboard mechanisms ensure strategic bidding is equivalent to truthful bidding.

Single-call design reduces computational overhead.

Mechanisms achieve desired allocations with minimal interaction.

Abstract

This paper gives a theoretical model for design and analysis of mechanisms for online marketplaces where a bidding dashboard enables the bid-optimization of long-lived agents. We assume that a good allocation algorithm exists when given the true values of the agents and we develop online winner-pays-bid and all-pay mechanisms that implement the same outcome of the algorithm with the aid of a bidding dashboard. The bidding dashboards that we develop work in conjunction with the mechanism to guarantee that bidding according to the dashboard is strategically equivalent (with vanishing utility difference) to bidding truthfully in the sequential truthful implementation of the allocation algorithm. Our dashboard mechanism makes only a single call to the allocation algorithm in each stage.

Equations

$$\tilde{p}(\mathsf{b})$$

$$\tilde{p}(\mathsf{v})$$

$$b(\mathsf{v})$$

$$\tilde{x}(\mathsf{b})$$

$$\mathsf{v}$$

$$c^{†}(\mathsf{v})$$

$$d(\hat{\mathsf{v}})$$

$$l(\hat{\mathsf{v}})$$

$${\mathsf{L}}^{(s+1)}={\mathsf{L}}^{(s)}+b^{(s)}({\hat{\mathsf{v}}}^{(s)})-{\mathsf{b}}^{(s)}.$$

$$\check{\mathsf{d}}$$

$${\check{\mathsf{L}}}^{(s+1)}={\check{\mathsf{L}}}^{(s)}+[b^{(s)}({\hat{\mathsf{v}}}^{(s)})-{\mathsf{b}}^{(s)}]{\check{\mathsf{x}}}^{(s)}.$$

$${\mathsf{L}}^{(t)}$$

$$\le \bar{\mathsf{d}}{\sum }_{k=0}^{\infty }(1-\eta {)}^k$$

$$=\bar{\mathsf{d}}/\eta .\text{qed}$$

$$y^{†}(\mathsf{z})$$

$${\sum }_{k=1}^{\check{\tau }}{c}^{(s_k)}(\mathsf{v})-{\sum }_{k=1}^{\check{\tau }}{b}^{(s_k)}(\mathsf{v})=b^{(s_0)}(\mathsf{v})-b^{(s_{\check{\tau }})}(\mathsf{v}).$$

$${\check{\mathsf{v}}}_i$$

$${\check{\mathsf{x}}}_i$$

$$\text{E}\left[{\check{\mathsf{r}}}_i\mid {\check{\mathsf{v}}}_{-i}\right]$$

$$={\mathsf{v}}_i(1-\rho )x_i({\mathsf{v}}_i,{\check{\mathsf{v}}}_{-i})-{\mathsf{v}}_i\frac{1-\rho }{\rho }\frac{\rho }{\bar{\mathsf{v}}}\frac{\bar{\mathsf{v}}}{{\mathsf{v}}_i}{\int }_0^{{\mathsf{v}}_i}{x}_i(z,{\check{\mathsf{v}}}_{-i})dz$$

$$=(1-\rho )p_i({\mathsf{v}}_i,{\check{\mathsf{v}}}_{-i}).$$

$${\check{\mathsf{L}}}^{(s+1)}={\check{\mathsf{L}}}^{(s)}+{\check{\mathsf{r}}}^{(s)}-{\check{\mathsf{w}}}^{(s)}{\mathsf{b}}^{(s)}.$$

$$\check{\mathsf{r}}-\check{\mathsf{w}}\mathsf{b}$$

$$\check{\mathsf{d}}$$

$$\text{E}\left[\check{\mathsf{d}}\right]$$

$${\check{\mathsf{L}}}^{(t)}-\text{E}\left[{\check{\mathsf{L}}}^{(t)}\right]=\sum _{k=0}^{\check{\tau }}\prod _{l=0}^{k-1}\left(1-\eta /y_i^{(s_l)}({\hat{\mathsf{v}}}^{(s_l)})\right)\left({\check{\mathsf{d}}}^{(s_k)}-\text{E}\left[{\check{\mathsf{d}}}^{(s_k)}\right]\right),$$

$$\displaystyle\text{\bf Pr}\!\left[{\left|{\check{{\mathsf{{{{\mathsf{L}}}}}}}}^{(t)}-\text{\bf E}\!\left[{{\check{{\mathsf{{{{\mathsf{L}}}}}}}}^{(t)}}\right]\right|>\xi\,\Big{|}\,{\check{{\mathsf{w}}}}^{(1)},\ldots,{\check{{\mathsf{w}}}}^{(t)}}\right]\leq 2\exp\left(-\frac{2\xi^{2}}{\sum\nolimits^{\infty}_{l=0}(1-{{{\mathsf{\eta}}}})^{2l}\bar{\mathrm{{{\mathsf{v}}}}}^{2}/\rho^{2}}\right)\leq 2\exp\left(-\frac{2\xi^{2}\rho^{2}{{{\mathsf{\eta}}}}}{\bar{\mathrm{{{\mathsf{v}}}}}^{2}}\right)$$

Full text

1 Introduction

This paper formalizes a theoretical study of bidding dashboards for the design of online markets. In these markets short-lived users arrive frequently and are matched with long-lived agents. Rarely in practice do market mechanisms, which allow the agents to bid to optimize their matching with the users, have truthtelling as an equilibrium strategy.222Such a matching is often the combination of a ranking of the agents by the market mechanism and the users’ choice behavior. Examples of the agents in such mechanisms include, sellers posting a buy-it-now price on eBay, hoteliers choosing a commission percentage on Booking.com, or advertisers bidding in Google’s ad auction. The first-two-examples are “winner pays bid” for the agents; none of these examples have truthtelling as an equilibrium. Though the Google Ad auction is not winner-pays-bid, its bidding dashboard displays clicks versus cost-per-click which is equivalent to a winner-pays-bid mechanism. The non-truthfulness of mechanisms for these markets is frequently the result of practical constraints; this paper is part of a small but growing literature on the design of these non-truthful mechanisms. By formally modeling bidding dashboards, which are common in these markets, this paper gives simple and practical approach to non-truthful mechanism design.

Two key challenges for the design of non-truthful mechanisms are (i) assisting the agents to find good strategies in these mechanisms and (ii) identifying mechanisms that have good outcomes when agents find good strategies. Bidding dashboards, which provide agents with market information relevant for bid optimization and are common in online marketplaces,333For example, Google provides a dashboard that forecasts click volume and cost per click as a function of advertisers’ bids and Booking.com provides a visibility booster that forecasts percentage of increased clicks relative to base commission as a function of the commission percentage offered by the hotel for bookings. can provide a solution to both issues. This paper gives a formal study of dashboard mechanisms and identifies dashboards and the accompanying mechanism that lead agents to find good strategies and in which good strategies result in good outcomes.

A novel technical feature of the mechanisms proposed by this paper is the use of econometric inference to infer the preferences of the agents in a way that permits the mechanism to then allocate nearly optimally for the inferred preferences. To illustrate this idea, consider the Nash equilibrium of the winner-pays-bid mechanism that allocates a single item to one of two agents with probability proportional to their bids, i.e., an agent wins with probability equal to her bid divided by the sum of bids, cf. the proportional share mechanism (e.g., Johari and Tsitsiklis,, 2004). Fixing the bid of agent 2, and assuming agent 1’s bid is in best response, the standard characterization of equilibria in auctions allows the inversion of agent 1’s bid-optimization problem and identification of agent 1’s value.444The bid-allocation rule of agent 1 is ${\tilde{{\mathnormal{x}}}}_{1}({{\mathsf{b}}})={{\mathsf{b}}}/({{\mathsf{b}}}+{{\mathsf{b}}}_{2})$. Agent 1 with value ${{\mathsf{v}}}_{1}$ chooses ${{\mathsf{b}}}_{1}=\operatorname{argmax}_{{{\mathsf{b}}}}({{\mathsf{v}}}_{1}-{{\mathsf{b}}})\,{\tilde{{\mathnormal{x}}}}_{1}({{\mathsf{b}}})$; this bid satisfies ${{\mathsf{v}}}_{1}={{\mathsf{b}}}_{1}+{\tilde{{\mathnormal{x}}}}_{1}({{\mathsf{b}}}_{1})/{\tilde{{\mathnormal{x}}}}_{1}^{\prime}({{\mathsf{b}}}_{1})=2{{\mathsf{b}}}_{1}+{{\mathsf{b}}}_{1}^{2}/{{\mathsf{b}}}_{2}$. For example, if ${{\mathsf{b}}}_{1}=1$ and ${{\mathsf{b}}}_{2}=2$, we can infer that agent 1’s value is ${{\mathsf{v}}}_{1}=5/2$. The same holds for agent 2. Thus, in hindsight the principal knows both agents’ values. The difficulty, which this paper addresses, of using this idea in a mechanism is the potential circular reasoning that results from attempting to allocate efficiently given these inferred values.

The paper considers general environments for single-dimensional agents, i.e., with preferences given by a value for service and utility given by value times service probability minus payment. The main goal of the paper is to convert an allocation algorithm, which maps the values of the agents to the set of agents that are served, into a mechanism that implements the same outcome but has a winner-pays-bid or all-pay payment format. In winner-pays-bid mechanisms the agents report bids, the mechanism selects a set of agents to win, and all winners pay their bids. In all-pay mechanisms the agents report bids, the mechanism selects a set of agents to win, and all agents pay their bids. For practical reasons, many markets are winner-pays-bid or all-pay.555All-pay mechanisms correspond to markets were agents pay monthly for service and the service level depends on the total payment which is fixed in advance by the agent’s bid.

From a theoretical point of view, the focus on winner-pays-bid and all-pay mechanisms is interesting because the prior literature has not identified good winner-pays-bid mechanisms for general environments (with the exception of Hartline and Taggart,, 2016, to be discussed with related work). For example, the natural highest-bids-win winner-pays-bid mechanism for the canonical environment of single-minded combinatorial auctions does not exhibit equilibria that are close to optimal (even for special cases where winner determination is computationally tractable; see Lucier and Borodin,, 2010, and Dütting and Kesselheim,, 2015).

The motivating marketplaces for this work are ones were short-lived users perform a search or collection of searches in order to identify one or more long-lived agents with whom to transact. The principal facilitates the user search by, for example, ranking the agents by a combination of perceived match quality for the user and the agents’ willingness to pay for the match, henceforth, the agents’ valuations. Strategically, the agents are in competition with each other for matching with the user. Viewing the user as a random draw from a population of users with varying tastes results in a stochastic allocation algorithm which maps the valuation profile of agents to the probabilities that a user selects each agent. This stochastic allocation algorithm is typically monotone in each agent’s value and continuous (e.g., Athey and Nekipelov,, 2010).

A dashboard gives the agent information about market conditions that enables bid optimization. Without loss in our setting, the dashboard is the agent’s predicted bid allocation rule. Our dashboard mechanism assumes that the bids are best response to the published dashboard, i.e., they “follow the dashboard”, and uses econometric inference to invert the profile of bids to get a profile of values. The dashboard mechanism for a given dashboard and allocation algorithm is as follows:

Publish the dashboard bid allocation rule to each agent and solicit each agent’s bid. 2. 1.

Invert the bid of each agent, assuming it is in best response to the published dashboard, to obtain an inferred value for each agent. 3. 2.

Execute the allocation algorithm on the inferred values to determine the outcome; charge each winner her bid (winner-pays-bid) or all agents their bids (all-pay).

Our main result is to identify dashboards where following the dashboard is not an assumption but near optimal strategy for the agents. For online marketplaces, we define winner-pays-bid and all-pay dashboards for any sequence of allocation algorithms where following the dashboard for a sequence of values is approximately strategically equivalent to bidding those values in the sequential truthful mechanism corresponding to the sequence of allocation algorithms. Thus, there is an approximate correspondence between all equilibria. The notion of approximate strategic equivalence is that the allocations are identical and the differences in payments vanish with the number of stages.

Our dashboards can be implemented in the (blackbox) single-call model for mechanism design. In this model, an algorithm has been developed that is monotonic in each agent’s value and obtains a good outcome. The mechanism’s only access to the allocation algorithm is by live execution, i.e., where the outcome of the algorithm is implemented. Babaioff et al., (2010) introduced this model and showed that truthful mechanisms can be single-call implemented. We show that winner-pays-bid and all-pay dashboard mechanisms can be single-call implemented in online marketplaces.

The above approach to the design of online markets is simple and practical. Though we have specified the mechanism and framework for winner-pays-bid and all-pay mechanisms and single-dimensional agents, the approach naturally generalizes to multi-dimensional environments and other kinds of mechanisms. It requires only that, from bids that are optimized for the dashboard, the preferences of the agents can be inferred. Moreover, payment formats can be mixed-and-matched across different agents within the same mechanism.

Related Work.

This paper is on non-revelation mechanism design as it applies to online markets. The goals of this paper – in providing foundations for non-revelation mechanism design – are closest to those of Hartline and Taggart, (2016, 2019). Both papers study iterated environments and the design of mechanisms with welfare that is arbitrarily close to the optimal welfare under general constraints. (Hartline and Taggart, additionally consider the objective of maximizing revenue.) The equilibrium concept of Hartline and Taggart, is Bayes-Nash equilibrium and it is assumed that agents are short-lived. These assumptions are not appropriate for the design of online markets considered in this paper, where agents are long-lived and possibly have persistent values.

There have been extensive studies of the price of anarchy of ad auctions, e.g., on the Google search engine (Leme and Tardos,, 2010; Caragiannis et al.,, 2011; Syrgkanis and Tardos,, 2013). A conclusion of these studies is that in standard models of equilibrium the industry-standard generalized second price auction can have welfare that is a constant factor worse than the optimal welfare. In these auctions, dashboards can also be used to infer agent values and the mapping from bids to values can be used in place of the “quality score” that is already in use in these mechanisms. Dashboard-based mechanisms like the ones we construct could obtain more efficient outcomes in these markets.

The approach of the main positive result of the paper is to link payments across successive stages in the sequential mechanism. Linking otherwise independent decisions has proven to be useful in overcoming difficulties in many areas of mechanism design. Jackson and Sonnenschein, (2007) show that linking decisions in i.i.d. sequential bilateral exchange can bypass the Myerson and Satterthwaite, (1983) impossibility result. Gorokh et al., (2017) shows that good mechanisms without money can be designed in a sequential environment by endowing agents with an artificial currency and running the sequential truthful mechanism with money. The endowment of artificial currency effectively links decisions across stages. Armstrong, (1999) shows that selling independent items in a bundle can raise higher revenue than selling separately, cf. Babaioff et al., (2014). The linking decisions idea is also in play in the previously discussed work on Bayes-Nash mechanism design by Hartline and Taggart, (2019).

The results of this paper can also be viewed as giving a blackbox reduction from truthful mechanism design to algorithm design. For single-dimensinal agents, a Bayes-Nash reduction from mechanism design to non-monotone algorithm design was given by Hartline and Lucier, (2010, 2015). Multi-dimensional reductions for addressing non-monotonicity were studied by Hartline et al., (2015); Bei and Huang, (2011); and Dughmi et al., (2017). Blackbox reductions of revenue maximization to welfare maximization were studied by Cai et al., 2013a ; Cai et al., 2013b . Single-call reductions, which reduce truthful mechanism design to algorithm design with a single blackbox call to the monotone algorithm were introduced by Babaioff et al., (2010) and further studied by Wilkens and Sivan, (2015). Our work extends this last area by giving single-call reductions to monotone algorithms for winner-pays-bid and all-pay mechanisms.

Organization.

The rest of this paper is organized as follows. Section 2 formalizes the multi-agent and sequential environment for mechanism design. Section 3 reviews single-agent auction theory. In particular, it shows that implementing winner-pays-bid and all-pay mechanisms for a single agent is straightforward. Section 4 gives the basic design of a dashboard mechanism. Section 5 gives a dashboard and proves that, in static environments, following the dashboard converges to Nash equilibrium and this equilibrium implements the algorithm’s desired allocation. Section 6 defines our dynamic solution concept for dashboard mechanisms, namely approximate strategic equivalence with the sequential truthful mechanism. In this section we describe a payment rebalancing dashboard that guarantees this approximate strategic equivalence. Section 7 shows that there are natural all-pay and winner-pays-bid dashboards for which following the dashboard with a static value results in low outstanding balance, even in a dynamic environment and without explicit rebalancing. Finally, Section 8 generalizes the dashboards to environments where the mechanism only has access to the allocation algorithm by single blackbox call.

2 The Sequential Multi-agent Model

This paper considers general environments for single-dimensional linear agents. There are $n$ agents $\{1,\ldots,n\}$. A profile of $n$ agent values is denoted $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}=({{\mathsf{v}}}_{1},\ldots,{{\mathsf{v}}}_{n})$. A multi-agent allocation algorithm is $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}:{\mathbb{R}}^{n}\to[0,1]^{n}$. We assume that any constraints of the environment are satisfied by the algorithm and will not otherwise be notating these constraints. A multi-agent non-truthful mechanism is denoted $(\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{x}}}}},\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{p}}}}})$ with bid allocation rule $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{x}}}}}:{\mathbb{R}}^{n}\to[0,1]^{n}$ and $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{p}}}}}:{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$. The agents have linear utility, for example, the utility of agent $i$ on bids $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{b}}}}$ in $(\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{x}}}}},\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{p}}}}})$ is ${{\mathsf{v}}}_{i}\,{\tilde{{\mathcal{x}}}}_{i}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{b}}}})-{\tilde{{\mathcal{p}}}}_{i}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{b}}}})$.

We consider agents interacting sequentially in a mechanism over many stages. For stage $s$ in $\{1,\ldots,t\}$, the environment is given by stage values $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}^{(s)}$ and a stage allocation algorithm $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}^{(s)}$. The designer chooses a stage mechanism $(\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{x}}}}}^{(s)},\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{p}}}}}^{(s)})$ and the agents choose stage bids $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{b}}}}^{(s)}$. The designer may choose the stage mechanism based on realized information and outcomes from previous stages.

We consider dynamic environments were the values and allocation algorithm can be distinct across stages and static environments where the values and the allocation algorithm are the same in every stage. In static environments both the designer may change the mechanism and the agents may change their bids from stage to stage.

An allocation algorithm takes several forms which will be notated distinctly. The ex post allocation algorithm is $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}:{\mathbb{R}}^{n}\to[0,1]^{n}$. The ex post allocation for agent $i$ is ${{{\mathcal{x}}}}_{i}:{\mathbb{R}}^{n}\to[0,1]$. Denote the profile with agent $i$’s value replaced with ${{{\mathsf{z}}}}$ is $({{{\mathsf{z}}}},\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}_{-i})=({{\mathsf{v}}}_{1},\ldots,{{\mathsf{v}}}_{i-1},{{{\mathsf{z}}}},{{\mathsf{v}}}_{i+1},\ldots,{{\mathsf{v}}}_{n})$. Fixing the other agent values $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}_{-i}$, the projection of the ex post allocation algorithm is single-agent allocation rule ${{{\mathnormal{x}}}}_{i}({{{\mathsf{z}}}})={{{\mathcal{x}}}}_{i}({{{\mathsf{z}}}},\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}_{-i})$. An allocation which could be output from $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}$ is $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{x}}}}$; agent $i$’s allocation is ${{\mathsf{x}}}_{i}$. Similar notation is adopted for other quantities that pertain to agents. For example the single-agent strategy function that maps agent $i$’s value to a bid is denoted by ${{{\mathnormal{b}}}}_{i}$ while a bid is denoted by ${{\mathsf{b}}}_{i}$.

3 Single-Agent Implementation

The multi-agent dashboard mechanisms of this paper will be based on a straightforward construction of winner-pays-bid and all-pay mechanisms for a single agent. The agent has a single-dimensional value ${{\mathsf{v}}}$ and linear utility ${{{\mathsf{u}}}}={{\mathsf{v}}}\,{{\mathsf{x}}}-{{\mathsf{p}}}$ for allocation probability ${{\mathsf{x}}}$ and expected payment ${{\mathsf{p}}}$. Given a stochastic allocation algorithm ${{{\mathnormal{x}}}}:{\mathbb{R}}\to[0,1]$, which specifies a desired mapping from the agent’s value to an allocation, we wish to implement ${{{\mathnormal{x}}}}$ when the agent behaves strategically in a mechanism with a winner-pays-bid or all-pay payment format.

A single-agent mechanism $({\tilde{{\mathnormal{x}}}},{\tilde{{\mathnormal{p}}}})$ maps a bid ${{\mathsf{b}}}$ to an allocation probability ${\tilde{{\mathnormal{x}}}}({{\mathsf{b}}})$ and payment ${\tilde{{\mathnormal{p}}}}({{\mathsf{b}}})$. Winner-pays-bid and all-pay mechanisms specify a bid allocation rule ${\tilde{{\mathnormal{x}}}}$ with bid payment rule ${\tilde{{\mathnormal{p}}}}$, respectively, defined as

[TABLE]

Strategic behavior of an agent in such a mechanism is governed by the $n=1$ agent special case of the characterization of Bayes-Nash equilibrium of Myerson, (1981). See Figure 1 for an illustration of the geometry of the payment identity.

Theorem 1** (A special case of Myerson,, 1981).**

Bidding strategy ${{{\mathnormal{b}}}}:{\mathbb{R}}\to{\mathbb{R}}$ is a best response for mechanism $({\tilde{{\mathnormal{x}}}},{\tilde{{\mathnormal{p}}}})$ if and only if its induced allocation and payment rules ${{{\mathnormal{x}}}}(\cdot)={\tilde{{\mathnormal{x}}}}({{{\mathnormal{b}}}}(\cdot))$ and ${{{\mathnormal{p}}}}(\cdot)={\tilde{{\mathnormal{x}}}}({{{\mathnormal{b}}}}(\cdot))$ satisfy:

1. (a)

monotonicity: ${{{\mathnormal{x}}}}(\cdot)$ is monotonically non-decreasing; 2. (b)

payment identity:

[TABLE]

and, for any value of the agent, any bid not in the range of the bidding strategy is dominated by a bid in its range.

A simple consequence of Theorem 1 is that with a monotone allocation algorithm ${{{\mathnormal{x}}}}$ the mechanism $({{{\mathnormal{x}}}},{{{\mathnormal{p}}}})$, with ${{{\mathnormal{p}}}}$ given by the payment identity, is truthful, i.e., bidding ${{{\mathnormal{b}}}}({{\mathsf{v}}})={{\mathsf{v}}}$ is an optimal strategy for the agent. We will refer to $({{{\mathnormal{x}}}},{{{\mathnormal{p}}}})$ with ${{{\mathnormal{p}}}}$ satisfying the payment identity as the truthful implementation of ${{{\mathnormal{x}}}}$. Often in the literature the payment that the agent makes with value ${{\mathsf{v}}}=0$ is ${{{\mathnormal{p}}}}(0)=0$.

The payment identity of Theorem 1 also allows the winner-pays-bid and all-pay mechanisms that implement a monotone allocation algorithm ${{{\mathnormal{x}}}}$ to be derived. In equations (3) and (4) these derivations are given with winner-pays-bid on the left and all-pay on the right. Specifically, the payment identity (2) and the formulae for payments from the payment format (1) give two equations for payments from which the bid strategy can be derived:

[TABLE]

Note that the winner-pays-bid and all-pay strategies ${{{\mathnormal{b}}}}$ from equation (3) as applied in equation (4) are distinct.

The above discussion assumes that we are given ${{{\mathnormal{x}}}}$ and which to construct a winner-pays-bid or all-pay mechanism $({\tilde{{\mathnormal{x}}}},{\tilde{{\mathnormal{p}}}})$. In this case, inferring values from bids can be accomplished with the inverse bidding strategy ${{{\mathnormal{b}}}}^{-1}$ from equation (3). If instead we were given $({\tilde{{\mathnormal{x}}}},{\tilde{{\mathnormal{p}}}})$ we could invert values from bids by evaluating the agent’s first order condition as follows. With value ${{\mathsf{v}}}$ the agent’s optimal bid ${{\mathsf{b}}}$ in single-agent mechanism $({\tilde{{\mathnormal{x}}}},{\tilde{{\mathnormal{p}}}})$ satisfies ${{\mathsf{v}}}={\tilde{{\mathnormal{p}}}}^{\prime}({{\mathsf{b}}})/{\tilde{{\mathnormal{x}}}}^{\prime}({{\mathsf{b}}})$. Evaluating this formula with the winner-pays-bid and all-pay payment formats of equation (1), respectively, we have:

[TABLE]

Thus, we see from any desired allocation rule ${{{\mathnormal{x}}}}$ there is a corresponding payment rule ${{{\mathnormal{p}}}}$, bid-allocation rule ${\tilde{{\mathnormal{x}}}}$, and bid strategy ${{{\mathnormal{b}}}}$. Unless explicitly stated otherwise, the payment rule, bid-allocation rule, and bid strategy correspond to transfer ${{{\mathnormal{p}}}}(0)=0$. In Section 4, we will define dashboards based on single-agent allocation rule ${{{\mathnormal{y}}}}$ with corresponding payment rule ${{{\mathnormal{q}}}}$, bid-allocation rule ${\tilde{{\mathnormal{y}}}}$, and bid strategy ${{{\mathnormal{c}}}}$. These notational conventions will be adopted throughout the paper. A key comparison will be of the desired allocation rule with ${{{\mathnormal{x}}}}$, ${{{\mathnormal{p}}}}$, ${\tilde{{\mathnormal{x}}}}$, and ${{{\mathnormal{b}}}}$ against the dashboard with ${{{\mathnormal{y}}}}$, ${{{\mathnormal{q}}}}$, ${\tilde{{\mathnormal{y}}}}$, and ${{{\mathnormal{c}}}}$.

In the above derivation of winner-pays-bid and all-pay implementations of the allocation rule ${{{\mathnormal{x}}}}$, it was assumed that the bid strategy ${{{\mathnormal{b}}}}$ is strictly increasing. When the allocation rule ${{{\mathnormal{x}}}}$ is strictly increasing and with ${{{\mathnormal{p}}}}(0)\leq 0$, ${{{\mathnormal{b}}}}$ is strictly increasing for both all-pay and winner-pays-bid implementations. When the transfer ${{{\mathnormal{p}}}}(0)>0$, is bid strategy ${{{\mathnormal{b}}}}$ is strictly increasing for all-pay implementations but not increasing for winner-pays-bid implementations.

Lemma 1**.**

For strictly increasing allocation rule ${{{\mathnormal{x}}}}$, all-pay bid functions ${{{\mathnormal{b}}}}$ are strictly increasing. For strictly increasing allocation rule ${{{\mathnormal{x}}}}$ and non-positive-transfer ${{{\mathnormal{p}}}}(0)\leq 0$, winner-pays-bid bid functions ${{{\mathnormal{b}}}}$ are strictly increasing.

Proof.

The all-pay result follows from equality of the bid function ${{{\mathnormal{b}}}}$ and payment rule ${{{\mathnormal{p}}}}$ and the strict monotonicity of the latter under the assumption that the allocation rule ${{{\mathnormal{x}}}}$ is strictly monotone. See Figure 1.

The winner-pays-bid result follows from differentiating the bid function and observing that the derivative is positive if and only if the surplus is at least the payment, i.e., ${{\mathsf{v}}}\,{{{\mathnormal{x}}}}({{\mathsf{v}}})>{{{\mathnormal{p}}}}({{\mathsf{v}}})$ for all ${{\mathsf{v}}}\geq 0$. Specifically, by the geometry of the payment identity (Figure 1), ${{{\mathnormal{p}}}}({{\mathsf{v}}})\leq 0$ implies ${{\mathsf{v}}}\,{{{\mathnormal{x}}}}({{\mathsf{v}}})>{{{\mathnormal{p}}}}({{\mathsf{v}}})$; while ${{{\mathnormal{p}}}}({{\mathsf{v}}})>0$ implies ${{\mathsf{v}}}\,{{{\mathnormal{x}}}}({{\mathsf{v}}})<{{{\mathnormal{p}}}}({{\mathsf{v}}})$ at ${{\mathsf{v}}}=0$. ∎

The dashboard mechanisms designed in the remainder of the paper effectively convert the mechanism design problem of implementing a multi-agent allocation algorithm into a collection of problems where the above single-agent derivations apply.

4 Dashboard Mechanisms

This section defines a family of dashboard mechanisms that give a practical approach to bidding and optimization in non-truthful Internet markets. The principal publishes a bidding dashboard which informs the agents of the market conditions; the agents use this dashboard to optimize their bids. We consider dashboards that give each agent estimates of the outcome of the mechanism for any possible bid of the agent. These estimates can be constructed, for example, from historical bid behavior of the agents in the mechanism.

Definition 1**.**

For a single-agent incentive compatible mechanism $({{{\mathnormal{y}}}},{{{\mathnormal{q}}}})$, the single-agent dashboard is ${\tilde{{\mathnormal{y}}}}:{\mathbb{R}}\to[0,1]$ as given by equation (4) and the bid-strategy for the dashboard is ${{{\mathnormal{c}}}}:{\mathbb{R}}\to{\mathbb{R}}$ as given by equation (3).666Thus, the notation for dashboard ${{{\mathnormal{y}}}}$ of payment rule ${{{\mathnormal{q}}}}$, bid-allocation rule ${\tilde{{\mathnormal{y}}}}$, and bid-strategy ${{{\mathnormal{c}}}}$ parallels the notation for allocation rule ${{{\mathnormal{x}}}}$ with ${{{\mathnormal{p}}}}$, ${\tilde{{\mathnormal{x}}}}$, and ${{{\mathnormal{b}}}}$. A multi-agent dashboard is a profile of single-agent bid-allocation rules $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathnormal{y}}}}}=({\tilde{{\mathnormal{y}}}}_{1},\ldots,{\tilde{{\mathnormal{y}}}}_{n})$ where ${\tilde{{\mathnormal{y}}}}_{i}({{\mathsf{b}}})$ is the forecast probability that agent $i$ wins with bid ${{\mathsf{b}}}$.

We consider the implementation of an allocation algorithm with a monotone allocation rule $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}:{\mathbb{R}}^{n}\to[0,1]^{n}$ by winner-pays-bid and all-pay dashboard mechanisms. The winner-pays-bid and all-pay format dashboards are identical except with respect to equations (3)-(5) which respectively define the value-to-bid optimal bidding function, the translation from (value) allocation rules to bid allocation rules, and the bid-to-value inversion for a bid-allocation rule.

We consider two design questions: (a) what dashboard should the principal publish and (b) what mechanism should the principal run. The goal is to pick a dashboard for which following the dashboard is a good strategy and if agents follow the dashboard then the allocation algorithm is implemented. In fact, if agents follow the dashboard, then the mechanism to implement the allocation algorithm is straightforward.

Definition 2**.**

The dashboard mechanism $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{x}}}}}:{\mathbb{R}}^{n}\to[0,1]^{n}$ for dashboard $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathnormal{y}}}}}$ and allocation algorithm $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}$ is:

Solicit bids $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{b}}}}$ for dashboard $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathnormal{y}}}}}$. 2. 1.

*Infer values $\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}$ from bids via equation (5). * 3. 2.

Output allocation $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{x}}}}}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{b}}}})=\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}(\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}})$ (with prices $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{p}}}}}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{b}}}})$ according to the payment format).

Notice that in the above definition the bid allocation rule $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathcal{x}}}}}$ is a mapping ${\mathbb{R}}^{n}\to[0,1]^{n}$ from bid profiles to allocation profiles while the dashboard $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathnormal{y}}}}}$ is a profile of single-agent bid allocation rules $({\mathbb{R}}\to[0,1])^{n}$ each of which maps a bid to an allocation probability. In other words, outcomes according to the dashboard are calculated independently across the agents while outcomes according to the mechanism depend on the reports of all agents together. (The notation distinguishes these kinds of allocation rules by using distinct typeface.) The following proposition follows simply from the correctness of equation (5).

Proposition 1**.**

In the dashboard mechanism (Definition 2) for any given strictly monotone and continuously differentiable dashboard and any allocation algorithm, if the agents follow the dashboard then the allocation algorithm is implemented.

5 Static Environments: Inferred Values Dashboards

Our analysis of dashboards in this section is restricted to static settings where both the agents’ values and the allocation algorithm do not change from stage to stage. How should the principal construct the dashboard to ensure the sequential dashboard mechanism, if all agents follow the dashboard, converges so that following the dashboard is a Nash equilibrium? For static environments, we will say that following the dashboard converges to Nash equilibrium if, assuming other agents follow the dashboard, an agent’s best response in the stage game converges to following the dashboard.

Our approach is motivated by the solution concept of fictitious play. In fictitious play agents best respond to the empirical distribution of the actions in previous rounds. Fictitious play assumes that the agents know the actions in past rounds. The principal could publish these actions; however, a better approach is to just publish as a dashboard the aggregate bid allocation rules that result. Our dashboard follows this approach except with respect to estimated values rather than actions. Following such a dashboard is, in a sense, an improvement on fictitious play.

Definition 3**.**

For stage $t+1\geq 2$, the inferred values dashboard is the profile of single-agent bid allocation rules $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathnormal{y}}}}}^{(t+1)}$ defined as follows:

The inferred valuation profile for stage $s\in\{1,\ldots,t\}$ is $\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}^{(s)}$. 2. 2.

The profile of single-agent allocation rules for stage $s\in\{1,\ldots,t\}$ is $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathnormal{x}}}}}^{(s)}$ with ${{{\mathnormal{x}}}}_{i}^{(s)}({{{\mathsf{z}}}})={{{\mathcal{x}}}}_{i}({{{\mathsf{z}}}},\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i}^{(s)})$ for each agent $i$. 3. 3.

The empirical profile of single-agent allocation rules at stage $t+1$ is $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathnormal{y}}}}}^{(t+1)}$ with ${{{\mathnormal{y}}}}_{i}^{(t+1)}({{{\mathsf{z}}}})=\frac{1}{t}\sum_{s\leq t}{{{\mathnormal{x}}}}_{i}^{(s)}({{{\mathsf{z}}}})$ for each agent $i$. 4. 4.

The inferred values dashboard is the profile of single-agent bid allocation rules $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathnormal{y}}}}}^{(t+1)}$ that correspond to profile $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathnormal{y}}}}}^{(t+1)}$ via equation (4).

Definition 4**.**

The $k$-lookback inferred values dashboard is the variant of the inferred values dashboard that averages over the last $\min(k,t)$ stages. The last-stage inferred values dashboard chooses $k=1$.

The inferred values dashboard does not specify a dashboard for stage

1. For stage 1, any strictly increasing dashboard will suffice. When the agents’ values and the allocation rule are static, following the dashboard converges to Nash equilibrium.

Theorem 2**.**

In static environments with fixed values $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}$ and continuous and strictly monotone stage allocation algorithm $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}$, in the sequential dashboard mechanism with the $k$-lookback inferred values dashboard, if agents follow the dashboard in stages $\max(t-k,1)$ through $t>1$ then following the dashboard in stage $t+1$ is a Nash equilibrium.

Proof.

The dashboards up to and including round $t$ are the average of the bid allocation rule for a continuous and strictly increasing allocation algorithm; thus they are continuous and strictly increasing. By equation (5) the inferred values of agents that follow the dashboard are the true values. These values are the same in each stage; thus the profile of single-agent allocation rules in each stage is the one that corresponds to the allocation algorithm on true valuation profile. The average of these profiles of allocation rule is the profile itself (they are all the same). The dashboard is the corresponding profile of single-agent bid allocation rules.

Consider agent $i$ and assume that other agents are following the dashboard in stage $t+1$. Thus, the estimated profile of other agent values is $\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i}^{(t+1)}=\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}_{-i}$. The allocation rule in value space faced by agent $i$ is ${{{\mathnormal{x}}}}_{i}^{(t+1)}({{{\mathsf{z}}}})={{{\mathcal{x}}}}_{i}({{{\mathsf{z}}}},\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}_{-i})$ which is equal to the allocation rule in all the previous stages. As the dashboard suggests bidding optimally according to the allocation rule of the previous stages, bidding according to the dashboard is agent $i$’s best response. Thus, following the dashboard is a Nash equilibrium in stage $t+1$. ∎

This dashboard mechanism is simple and practical and can implement any strictly monotone and continuously differentiable allocation algorithm. For example, welfare maximization with a convex regularizer gives such a allocation algorithm. For the paradigmatic problem of single-minded combinatorial auction environments, our dashboard mechanism with the regularized welfare-maximization allocation algorithm gives outcomes that can be arbitrarily close to optimal (for the appropriate regularizer).

A critical issue with the guarantee of Theorem 2 is that it is delicate to the Nash assumption. If other agents do not follow the dashboard, then following the dashboard is not necessarily a good strategy for the agent. The remainder of the paper will resolve this issue by giving stronger analyses and stronger dashboards. Specifically, we will give a dashboard for which the sequential dashboard mechanism is strategically equivalent to the sequential truthful mechanism.

6 Dynamic Environments: Payment Rebalancing Dashboards

In this section we develop a dashboard that will satisfy the strong guarantee that the $\epsilon$ equilibria of the sequential dashboard mechanism are approximately the same as the $\epsilon$ equilibria from running the truthful mechanism that implements the allocation algorithm in each stage. Moreover, the follow-the-dashboard strategy corresponds to the truthtelling strategy and, thus, is an $\epsilon$-equilibrium.777Note that the sequential truthful mechanism will generally have many other equilibria as well.

Definition 5**.**

Two sequential mechanisms are $\epsilon$ approximately strategically equivalent if for all sequences of bids profiles in one mechanism there is a corresponding sequence of bid profiles in the other mechanism such that the absolute per-stage-average difference in utilities of any agent in the two mechanisms is at most $\epsilon$, and vice versa.

Definition 6**.**

For a sequence of monotone stage allocation algorithms $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}^{(1)},\ldots,\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}^{(t)}$, the sequential truthful mechanism is given by the stage mechanism $(\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}^{(s)},\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{p}}}}}^{(s)})$ for stage $s\in\{1,\ldots,t\}$ where ${{{\mathcal{p}}}}_{i}^{(s)}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}})$ is defined from the payment identity applied to ${{{\mathcal{x}}}}_{i}(\cdot,\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}_{-i})$ for each agent $i$.

Definition 7**.**

A sequential dashboard mechanism is $\epsilon$ incentive consistent if it is $\epsilon$ strategically equivalent to the sequential truthful mechanism and the follow-the-dashboard strategy corresponds to the truthtelling strategy.

We now develop a dashboard for which the sequential dashboard mechanism is $\epsilon$ incentive consistent with $\epsilon$ that vanishes with the number of stages. The high-level approach of this dashboard is simple: In any stage where the actual payment and the truthful payment are different, add the payment residual to a balance and adjust future dashboards to either collect additional payment or to discount the required payment so that the outstanding balance is resolved over a few subsequent stages. The difference in utilities of an agent in such a dashboard mechanism and the truthful mechanism with the same allocation algorithm is bounded by the per-stage payment residual and the number of stages it takes to resolve it. When these quantities are both constants, the average per-stage difference between the dashboard mechanism’s outcome and the truthful mechanism’s outcome vanishes with the number of stages.

The existence of this dashboard shows that in a repeated scenario there is essentially no difference between winner-pays-bid, all-pay, and truthful payment formats. In some sense, linking payments between stages and adjusting the mapping from bids to values allows a market designer to choose a payment format that is appropriate for the application.

Three final comments before presenting the mechanism: First, balancing payments in all-pay mechanisms is much easier than balancing payments in winner-pays-bids mechanisms. The reason is simple, in all-pay mechanisms the payment is deterministic and, thus, any additional payment requested is paid exactly. On the other hand, payments collected in winner-pays-bid mechanisms depend on the probability of allocation. If this probability of allocation is fluctuating then collected payments can over- or under-shoot a target. We give an approach below that resolves this issue. Second, in the static setting where the agent values and the allocation algorithm are unchanging, in the follow-the-dashboard equilibrium of the inferred values dashboard mechanism (Definition 3) the only non-trivial payment residual is in the first stage, once this balance is resolved, the accrual of subsequent balance is off the equilibrium path. Thus, in steady state the rebalancing required by the dashboard is trivial. Third, dashboards with payment rebalancing allow dynamically changing environments, e.g., agent values and the allocation algorithm. For agents that follow the dashboard, the rebalancing mechanism only kicks in when the agent’s value or the environment changes.

The main definition and proposition that motivate the approach are as follows.

Definition 8**.**

Define the outstanding balance of an agent in a $t$-stage sequential dashboard mechanism for inferred values $\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}^{(1)},\ldots,\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}^{(t)}$ as the magnitude of the total expected difference in payments between the sequential dashboard mechanism and the sequential truthful mechanism when agents report values $\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}^{(1)},\ldots,\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}^{(t)}$.

Proposition 2**.**

A $t$-stage dashboard mechanism with worst-case (over valuation profiles) outstanding balance upper bounded by $\epsilon t$ is $\epsilon$ incentive consistent.

Proof.

Fix $t$ valuation profiles $\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}^{(1)},\ldots,\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}^{(t)}$ and consider the allocation and payments from the sequential dashboard mechanism where agents follow the dashboard for these valuation profiles and the sequential truthful mechanism where agents report these valuation profiles. The allocations obtained by these two mechanisms are identical as both call the stage $s$ allocation algorithm $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}^{(s)}$ on valuation profile $\kern 0.43057pt{\hat{{\kern-0.43057pt\bm{\mathsf{v}}}}}^{(s)}$. By the assumption of the proposition, the outstanding balance of any agent, i.e., the total difference in payments of the mechanisms, is upper bounded by $\epsilon t$. For any true valuation profiles, $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}^{(1)},\ldots,\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}^{(t)}$, and because the utilities of the agent are linear in payments; the per-stage-average difference in utility between the two mechanisms is at most $\epsilon$. Thus, the definition of $\epsilon$ incentive consistency is satisfied. ∎

6.1 Rebalancing Dashboards

In the subsequent developments we will focus on a single agent. We are given a dashboard with alloction rule ${{{\mathnormal{y}}}}$ (and corresponding payment rule ${{{\mathnormal{q}}}}$, bid allocation rule ${\tilde{{\mathnormal{y}}}}$, and bid strategy ${{{\mathnormal{c}}}}$). The rebalancing method will construct a new dashboard that corresponds to ${{{\mathnormal{y}}}}^{\dagger}$, ${{{\mathnormal{q}}}}^{\dagger}$, ${\tilde{{\mathnormal{y}}}}^{\dagger}$, and ${{{\mathnormal{c}}}}^{\dagger}$. The general approach of the construction is that, if the allocation algorithm and dashboard correspond with ${{{\mathnormal{x}}}}={{{\mathnormal{y}}}}$ then the dashboard is correct and the correct payments are made, and otherwise, the dashboard is incorrect and the incorrect payments will need to be resolved by future dashboards.

The main idea in the proposed rebalancing approach is that adding a constant to the agent’s expected payment, i.e., setting transfer ${{{\mathnormal{q}}}}(0)\neq 0$ in the payment identity, does not affect incentives. Unfortunately, there are two potential issues with this approach. Crucially, by Lemma 1, winner-pay-bid bid functions are not invertible for ${{{\mathnormal{q}}}}(0)>0$. Thus, such a dashboard mechanism cannot be implemented as described thus far. Moreover, with ${{{\mathnormal{q}}}}(0)>0$ the stage mechanism is not individually rational. This latter concern is not as serious as the former concern as an agent’s long run utility will be non-negative from participating even when the agent’s utility in a single stage could be negative. Our approach will be to view these dashboards as a reference for subsequently developed dashboards which satisfy ${{{\mathnormal{q}}}}(0)=0$ and both admit invertible bid functions and satisfy individual rationality.

Definition 9**.**

For a single-agent truthful mechanism $({{{\mathnormal{y}}}},{{{\mathnormal{q}}}})$, the single-agent reference rebalancing dashboard for rebalancing rate ${{{\mathsf{\eta}}}}\in(0,1]$ and outstanding balance ${{{\mathsf{L}}}}$ is ${\tilde{{\mathnormal{y}}}}^{\dagger}$ that corresponds to payment rule ${{{\mathnormal{q}}}}^{\dagger}$ defined as ${{{\mathnormal{q}}}}^{\dagger}({{\mathsf{v}}})={{{\mathnormal{q}}}}({{\mathsf{v}}})+{{{\mathsf{L}}}}\,{{{\mathsf{\eta}}}}$, i.e., with ${{{\mathnormal{q}}}}^{\dagger}(0)={{{\mathsf{L}}}}\,{{{\mathsf{\eta}}}}$.

From Definition 9 and equation (3), the bidding strategy can be calculated for winner-pays-bid and all-pay dashboards, respectively

[TABLE]

The bid-allocation rule of the dashboard ${\tilde{{\mathnormal{y}}}}^{\dagger}$ is then defined via the allocation rule ${{{\mathnormal{y}}}}$ and the inverse of the strategy ${{{\mathnormal{c}}}}^{\dagger}$ via equation (4). The bid of an agent should be viewed as shown in equation (6) as two terms. The first term is for the original dashboard ${{{\mathnormal{y}}}}$ and the second term is for resolving the outstanding balance. After each stage the balance is adjusted to account for how much of the outstanding balance was resolved and by any new payment residual resulting from misestimation of the dashboard ${{{\mathnormal{y}}}}$ for the realized allocation rule ${{{\mathnormal{x}}}}$. While our analysis will keep track of when payment residuals are generated and how long it takes to resolve them, the balance tracking need only consider the difference between what was paid and what should have been paid for the realized allocation rule.

Definition 10**.**

For allocation rule ${{{\mathnormal{x}}}}$ with bid-strategy ${{{\mathnormal{b}}}}$, dashboard allocation rule ${{{\mathnormal{y}}}}$, and inferred value ${\hat{{\mathsf{v}}}}$, the payment residual for the reference rebalancing dashboard ${\tilde{{\mathnormal{x}}}}^{\dagger}$ with realized allocation rule ${{{\mathnormal{x}}}}$ is

[TABLE]

respectively. The total change to the balance is ${{{\mathnormal{d}}}}({\hat{{\mathsf{v}}}})-{{{\mathnormal{l}}}}({\hat{{\mathsf{v}}}})$.

Note that in the above definition for winner-pays-bid mechanisms, if ${{{\mathnormal{x}}}}={{{\mathnormal{y}}}}$ then ${{{\mathnormal{b}}}}={{{\mathnormal{c}}}}$ and ${{{\mathnormal{d}}}}({\hat{{\mathsf{v}}}})=0$. For all-pay mechanisms with ${{{\mathnormal{x}}}}={{{\mathnormal{y}}}}$, then ${{{\mathnormal{b}}}}={{{\mathnormal{c}}}}$ and ${{{\mathnormal{d}}}}({\hat{{\mathsf{v}}}})=0$. The perspective to have is that in steady-state it should be that ${{{\mathnormal{x}}}}={{{\mathnormal{y}}}}$ and there should be no payment residual but, when agents arrive, depart, or have value changes, then there can be non-trivial payment residual which will be rebalanced by the dashboard. More generally the following lemma bounds the magnitude of the payment residual.

Lemma 2**.**

In a stage in which the agent’s inferred value is ${\hat{{\mathsf{v}}}}$, the magnitude of payment residual $|{{{\mathnormal{d}}}}({\hat{{\mathsf{v}}}})|$ is at most ${\hat{{\mathsf{v}}}}$.

Proof.

From equation (7), for all-pay mechanisms the payment residual is the difference in bids in an individually rational mechanism. As bids for an agent with value ${\hat{{\mathsf{v}}}}$ are between [math] and ${\hat{{\mathsf{v}}}}$, the magnitude of their difference is at most ${\hat{{\mathsf{v}}}}$. For winner-pays-bid ${{{\mathnormal{b}}}}({\hat{{\mathsf{v}}}})-{{{\mathnormal{c}}}}({\hat{{\mathsf{v}}}})$ is again a difference of bids and multiplying this difference by ${{{\mathnormal{x}}}}({\hat{{\mathsf{v}}}})\in[0,1]$ preserves the bound of ${\hat{{\mathsf{v}}}}$ on its magnitude. ∎

The analysis in the sections below makes no assumption about whether the mechanism is in steady-state. Both the allocation algorithm and the agents’ values may change from stage to stage. The main idea of the analysis is to consider the payment residual ${{{\mathnormal{d}}}}^{(s)}$ in stage $s$ and calculate how much of it can remain after stage $t$ when there is a guaranteed fraction of it is rebalanced as part of ${{{\mathnormal{l}}}}^{(s^{\prime})}$ for subsequent stages $s^{\prime}\in\{s+1,\ldots,t\}$. We give a worst case analysis that pessimisticly assumes that the payment residual in each stage is the same sign and equal to its maximum value, i.e., the value of the agent. These bounds apply to the rebalancing dashboard constructed from any original dashboard that may not have any relation to the realized allocation rule.

6.2 Outstanding Balance of the All-pay Rebalancing Dashboard

Definition 11**.**

From stage $s$ with outstanding balance ${{{\mathsf{L}}}}^{(s)}$, agent bid ${{\mathsf{b}}}^{(s)}$, inferred value ${\hat{{\mathsf{v}}}}^{(s)}$, realized allocation rule ${{{\mathnormal{x}}}}^{(s)}$, and corresponding all-pay bid strategy ${{{\mathnormal{b}}}}^{(s)}$; the stage $s+1$ outstanding balance in the all-pay rebalancing dashboard is:

[TABLE]

The analysis of the rebalancing dashboard for all-pay mechanisms is trivial. To parallel the subsequent presentation for winner-pays-bid mechanisms below, we write the formal lemma and theorem.

Lemma 3**.**

In the all-pay dashboard with rebalancing rate ${{{\mathsf{\eta}}}}\in(0,1]$, in any stage the balance resolved is ${{{\mathnormal{l}}}}({\hat{{\mathsf{v}}}})={{{\mathsf{L}}}}\,{{{\mathsf{\eta}}}}$; for ${{{\mathsf{\eta}}}}=1$ the full outstanding balance is resolved.

Proof.

See equation (8). ∎

Theorem 3**.**

In dynamic environments, the sequential dashboard mechanism with an all-pay rebalancing dashboard with rebalancing rate ${{{\mathsf{\eta}}}}=1$ and per-stage estimated value at most $\bar{\mathrm{{{\mathsf{v}}}}}$; the outstanding balance at stage $t$ is at most $\bar{\mathrm{{{\mathsf{v}}}}}$ and, consequently, over $t$ stages the dashboard mechanism is $\bar{\mathrm{{{\mathsf{v}}}}}/t$ incentive consistent.

Proof.

Lemma 2 upper bounds the payment residual of each stage by $\bar{\mathrm{{{\mathsf{v}}}}}$ (which is added to the outstanding balance). Lemma 3 upper bounds the outstanding balance prior to stage $s$ that is resolved in stage $s$. With ${{{\mathsf{\eta}}}}=1$ the outstanding balance at stage $t$ is only the payment residual from stage $t$. Thus, the outstanding balance after stage $t$ is upper bounded by $\bar{\mathrm{{{\mathsf{v}}}}}$. Applying 2 the sequential dashboard mechanism is $\bar{\mathrm{{{\mathsf{v}}}}}/t$ incentive consistent. ∎

Note that Theorem 3 can be improved in static environments where both inferred stage values ${\hat{{\mathsf{v}}}}^{(s)}$ and the single-agent allocation rules induced from the values of the other agents ${{{\mathnormal{x}}}}^{(s)}$ are static, i.e., ${\hat{{\mathsf{v}}}}^{(s)}={\hat{{\mathsf{v}}}}$ and ${{{\mathnormal{x}}}}^{(s)}={{{\mathnormal{x}}}}$. In these cases the inferred values dashboard (Definition 3) is ${{{\mathnormal{y}}}}^{(s)}={{{\mathnormal{x}}}}$ in all stages but the first and only this first stage accrues payment residual. Thus, the total balance at stage $t\geq 2$ with ${{{\mathsf{\eta}}}}=1$ is ${{{\mathsf{L}}}}^{(t)}=0$.

6.3 Outstanding Balance of the Winner-pays-bid Rebalancing Dashboard

Defining good rebalancing dashboards is more challenging for winner-pays-bid mechanisms as the balance resolved depends on the dashboard at the estimated value ${{{\mathnormal{y}}}}({\hat{{\mathsf{v}}}})$ which we do not require to be constant across rounds. We therefore chose a conservative rebalancing rate ${{{\mathsf{\eta}}}}<1$ to avoid overshooting the target of zero balance. Specifically, we will set ${{{\mathsf{\eta}}}}$ less than the minimum allocation probability of the dashboard; thus ${{{\mathsf{\eta}}}}/{{{\mathnormal{y}}}}({\hat{{\mathsf{v}}}})\leq 1$ in equation (8). As the rebalancing dashboard can fix any incorrect dashboard, it may be desirable to distort the dashboard at the low end to guarantee that that the minimum allocation probability of the dashboard is not too small.

Since we are considering winner-pays-bid mechanisms we will only adjust an agent’s balance when the agent is allocated. To make explicit the actual values versus their expectations we will adopt notation ${\check{{\mathsf{{{{\mathsf{L}}}}}}}}$ and ${\check{{\mathsf{x}}}}$ for the actual balance and actual allocation with ${{{\mathsf{L}}}}=\text{\bf E}\!\left[{{\check{{\mathsf{{{{\mathsf{L}}}}}}}}}\right]$ and ${{{\mathnormal{x}}}}({\hat{{\mathsf{v}}}})=\text{\bf E}\!\left[{{\check{{\mathsf{x}}}}}\right]$. We rewrite, per the discussion above, the payment residual and balanced resolved from equations (7) and (8) for inferred value ${\hat{{\mathsf{v}}}}$ as:

[TABLE]

where $\text{\bf E}\!\left[{{\check{{\mathsf{d}}}}}\right]={{{\mathnormal{d}}}}({\hat{{\mathsf{v}}}})$ and $\text{\bf E}\!\left[{{\check{{\mathsf{{{{\mathnormal{l}}}}}}}}}\right]={{{\mathnormal{l}}}}({\hat{{\mathsf{v}}}})$. Note that ${{{\mathnormal{b}}}}$ is the winner-pays-bid bid strategy corresponding to ${{{\mathnormal{x}}}}$ while ${{{\mathnormal{c}}}}$ is the winner-pays-bid bid strategy for dashboard ${{{\mathnormal{y}}}}$.

Definition 12**.**

From stage $s$ with outstanding balance ${{{\mathsf{L}}}}^{(s)}$, agent bid ${{\mathsf{b}}}^{(s)}$, inferred value ${\hat{{\mathsf{v}}}}^{(s)}$, realized allocation rule ${{{\mathnormal{x}}}}^{(s)}$, corresponding winner-pays-bid bid strategy ${{{\mathnormal{b}}}}^{(s)}$, and realized allocation ${\check{{\mathsf{x}}}}^{(s)}$; the stage $s+1$ outstanding balance in the all-pay rebalancing dashboard is:

[TABLE]

Denote by $\check{T}=\{s:{\check{{\mathsf{x}}}}^{(s)}\neq 0\}$ the set of stages where the agent is allocated and by $\check{\tau}={\left|\check{T}\right|}$ the number of such stages. These are the stages where ${\check{{\mathsf{d}}}}^{(s)}$ and ${\check{{\mathsf{{{{\mathnormal{l}}}}}}}}^{(s)}$ can be non-zero. We consider the amount of the payment residual ${\check{{\mathsf{d}}}}^{(s)}$ from stage $s\in\check{T}$ that remains at final stage $t$ when, at each subsequent stage in $\check{T}$, a fraction of that payment residual is resolved.

Lemma 4**.**

At any rebalancing stage $s\in\check{T}$ and under any agent strategy, the winner-pays-bid rebalancing dashboard with rebalancing rate ${{{\mathsf{\eta}}}}\in(0,1)$ for dashboard allocation rule ${{{\mathnormal{y}}}}^{(s)}$ with allocation probability supported on $[{{{\mathsf{\eta}}}},1]$ resolves balance ${\check{{\mathsf{{{{\mathnormal{l}}}}}}}}$ between ${\check{{\mathsf{{{{\mathsf{L}}}}}}}}\,{{{\mathsf{\eta}}}}$ and ${\check{{\mathsf{{{{\mathsf{L}}}}}}}}$.

Proof.

This result follows from the definition of ${\check{{\mathsf{{{{\mathnormal{l}}}}}}}}$ and the fact that ${{{\mathsf{\eta}}}}\leq{{{\mathsf{\eta}}}}/{{{\mathnormal{y}}}}({\hat{{\mathsf{v}}}})\leq 1$ for all ${\hat{{\mathsf{v}}}}$ by the assumption that $1\geq{{{\mathnormal{y}}}}({\hat{{\mathsf{v}}}})\geq{{{\mathsf{\eta}}}}$. ∎

The following theorem about the winner-pays-bid dashboard rebalancing mechanism will upper bound the outstanding balance at any time. With these quantities taken as constants relative to the number of stages $t$, the imbalance per stage is vanishing with $t$. It is useful to contrast the assumptions and bounds of the analogous result for all-pay dashboards (Theorem 3) with Theorem 4.

Theorem 4**.**

For any monotonic stage allocation rules ${{{\mathnormal{x}}}}^{(1)},\ldots,{{{\mathnormal{x}}}}^{(t)}$ and any monotonic dashboard rules ${{{\mathnormal{y}}}}^{(1)},\ldots,{{{\mathnormal{y}}}}^{(t)}$ with probabilities supported on $[{{{\mathsf{\eta}}}},1]$, the winner-pays-bid payment rebalancing dashboard with rebalancing rate ${{{\mathsf{\eta}}}}$ and per-stage payment residual at most ${\bar{{\mathsf{d}}}}$ has outstanding balance at stage $t$ of at most ${\bar{{\mathsf{d}}}}/{{{\mathsf{\eta}}}}$.

Proof.

If we start a stage $s$ with outstanding balance ${\check{{\mathsf{{{{\mathsf{L}}}}}}}}^{(s)}$, Lemma 4 implies that at least a ${{{\mathsf{\eta}}}}$ (and at most 1) fraction of it is resolved. The balance remaining is at most $(1-{{{\mathsf{\eta}}}})\,{\check{{\mathsf{{{{\mathsf{L}}}}}}}}^{(s)}$ (and at least zero).

Denote the stages that the agent is allocated, indexed in decreasing order, by $\check{T}=\{s_{0},\ldots,s_{\check{\tau}-1}\}$. The payment residual ${\check{{\mathsf{d}}}}^{(s_{k})}$ from stage $s_{k}$ that remains in the final stage $t$ is at most $(1-{{{\mathsf{\eta}}}})^{k}\,{\check{{\mathsf{d}}}}^{(s_{k})}$. (This bound is worst case, specifically, we do not track the possibility that some of the balance might cancel with new payment residual of the opposite sign.) Summing over each stage $s_{k}\in\check{T}$, we can bound the outstanding balance at stage $t$ by:

[TABLE]

With an upper bound on the per-stage estimated value of $\bar{\mathrm{{{\mathsf{v}}}}}$, Theorem 4 can be combined with Lemma 2, which bounds ${\bar{{\mathsf{d}}}}\leq\bar{\mathrm{{{\mathsf{v}}}}}$, to obtain the following corollary. The sequential dashboard mechanism with a rebalancing dashboard is $\epsilon$ incentive consistent with $\epsilon\leq\bar{\mathrm{{{\mathsf{v}}}}}/({{{\mathsf{\eta}}}}\,t)$ in $t$ rounds (see Definition 7 and 2). With $\bar{\mathrm{{{\mathsf{v}}}}}/{{{\mathsf{\eta}}}}$ held as constants the incentive inconsistency vanishes with $t$.

Corollary 1**.**

In dynamic environments, the sequential dashboard mechanism with a winner-pays-bid rebalancing dashboard with rebalancing rate ${{{\mathsf{\eta}}}}\in(0,1)$, dashboard allocation probabilities supported on $[{{{\mathsf{\eta}}}},1]$, and per-stage estimated value at most $\bar{\mathrm{{{\mathsf{v}}}}}$; the outstanding balance at stage $t$ is at most $\bar{\mathrm{{{\mathsf{v}}}}}/{{{\mathsf{\eta}}}}$ and, consequently, over $t$ stages the dashboard mechanism is $\bar{\mathrm{{{\mathsf{v}}}}}/({{{\mathsf{\eta}}}}\,t)$ incentive consistent.

Note that Theorem 4 can be improved in static environments where both inferred stage values ${\hat{{\mathsf{v}}}}^{(s)}$ and the single-agent allocation rules induced from the values of the other agents ${{{\mathnormal{x}}}}^{(s)}$ are static, i.e., ${\hat{{\mathsf{v}}}}^{(s)}={\hat{{\mathsf{v}}}}$ and ${{{\mathnormal{x}}}}^{(s)}={{{\mathnormal{x}}}}$. In these cases the inferred values dashboard (Definition 3) is ${{{\mathnormal{y}}}}^{(s)}={{{\mathnormal{x}}}}$ in all stages but the first and only this stage accrues non-trivial payment residual. In this case, the total balance at stage $t$ is ${{{\mathsf{L}}}}^{(t)}\leq{\hat{{\mathsf{v}}}}\,(1-{{{\mathsf{\eta}}}})^{\check{\tau}-1}$, i.e., exponentially small in the number of stages that the agent is allocated.

6.4 Individual Rationality and Non-negative Transfers

The rebalancing dashboards presented and analyzed above set a transfer ${{{\mathnormal{q}}}}(0)\neq 0$ so that payment residuals of early stages are resolved in subsequent stages. Individual rationality requires that ${{{\mathnormal{q}}}}(0)\leq 0$ and must be satisfied for winner-pays-bid bidding strategies to be invertible. Non-negative transfers requires that ${{{\mathnormal{q}}}}(0)\geq 0$. Typical mechanisms satisfy both properties, i.e., ${{{\mathnormal{q}}}}(0)=0$.

In this section we describe winner-pays-bid dashboards with transfer ${{{\mathnormal{q}}}}(0)=0$; the main idea is for the dashboard outcome to look to high-valued agents like ${{{\mathnormal{q}}}}(0)\neq 0$ but, for low-valued agents where such an outcome is not feasible with ${{{\mathnormal{q}}}}(0)=0$, to have payment as close as possible to the ${{{\mathnormal{q}}}}(0)=0$ outcome. Notice that the highest individually rational payment for an agent with value ${{\mathsf{v}}}$ is ${{\mathsf{v}}}$; the lowest non-negative payment for an agent is 0. Thus, we seek to identify allocation rules with approximately these payments for low-valued agents to combine with the original allocation rule for high-valued agents.

The following observation which follows from the derivation (3) of the winner-pays-bid bid strategy of ${{{\mathnormal{c}}}}({{{\mathsf{z}}}})={{{\mathnormal{q}}}}({{{\mathsf{z}}}})/{{{\mathnormal{y}}}}(z)$ motivates the developments of this section. Specifically, scaling the dashboard allocation ${{{\mathnormal{y}}}}$ by some factor $\alpha>0$ similarly scales the payments ${{{\mathnormal{q}}}}$ by the same factor; the bid function is their ratio which is constant with respect to this scaling.

Fact 1**.**

Winner-pays-bid bid strategies are independent of the scale of the allocation rule, i.e., the bid strategies ${{{\mathnormal{c}}}}$ and ${{{\mathnormal{c}}}}^{\dagger}$ corresponding to ${{{\mathnormal{y}}}}$ and ${{{\mathnormal{y}}}}^{\dagger}$, defined as ${{{\mathnormal{y}}}}^{\dagger}({{{\mathsf{z}}}})=\alpha\,{{{\mathnormal{y}}}}({{{\mathsf{z}}}})$ for any $\alpha>0$, are identical.

We define a family of allocation rules which admit simple linear bid strategies. An allocation rule in the family can be selected to require either a very high bid or a very low bid, depending on whether we need to rebalance a positive or negative balance.

Definition 13**.**

The $\gamma\in(0,1)$ linear-bid allocation rule is ${{{\mathnormal{y}}}}^{\gamma}({{{\mathsf{z}}}})={{{\mathsf{z}}}}^{\nicefrac{{\gamma}}{{1-\gamma}}}$; the linear-bid allocation rule that meets point $({{\mathsf{v}}},{{\mathsf{x}}})$ is ${{{\mathnormal{y}}}}^{\gamma,{{\mathsf{v}}},{{\mathsf{x}}}}({{{\mathsf{z}}}})=[\nicefrac{{{{\mathsf{x}}}}}{{{{{\mathnormal{y}}}}^{\gamma}({{\mathsf{v}}})}}]\,{{{\mathnormal{y}}}}^{\gamma}({{{\mathsf{z}}}})$.

Lemma 5**.**

The bid strategy for the $\gamma$ linear-bid allocation rule ${{{\mathnormal{y}}}}^{\gamma,{{\mathsf{v}}},{{\mathsf{x}}}}$ is ${{{\mathnormal{c}}}}^{\gamma,{{\mathsf{v}}},{{\mathsf{x}}}}({{{\mathsf{z}}}})={{{\mathnormal{c}}}}^{\gamma}({{{\mathsf{z}}}})=\gamma\,{{{\mathsf{z}}}}$.

Proof.

The proof follows by calculation of the bid function according to equation (3). By 1, it suffices to calculate the bid function for ${{{\mathnormal{y}}}}^{\gamma}$. ∎

The payment rebalancing dashboard is constructed from an original dashboard by replacing its allocation to low values with an appropriately scaled linear-bid allocation rule.

Definition 14**.**

The payment rebalancing dashboard ${{{\mathnormal{y}}}}^{\dagger}$ for outstanding balance ${{{\mathsf{L}}}}$ and original dashboard ${{{\mathnormal{y}}}}$ is constructed as follows.888Notes: The desired ${{\mathsf{v}}}^{\dagger}$ exists as ${{{\mathnormal{c}}}}$ is continuously increasing but finite. In the case where ${{{\mathsf{L}}}}=0$, the definition sets ${{\mathsf{v}}}^{\dagger}=0$ and ${{{\mathnormal{y}}}}^{\dagger}={{{\mathnormal{y}}}}$.

For non-negative balance ${{{\mathsf{L}}}}\geq 0$, set $\gamma\to 1$, i.e., just below 1; and set ${{\mathsf{v}}}^{\dagger}$ to solve ${{\mathsf{v}}}^{\dagger}-{{{\mathnormal{c}}}}({{\mathsf{v}}}^{\dagger})={{{\mathsf{L}}}}$. 2. 2.

For negative balance ${{{\mathsf{L}}}}<0$, set $\gamma\to 0$, i.e., just above 0; and set ${{\mathsf{v}}}^{\dagger}$ to solve $-{{{\mathnormal{c}}}}({{\mathsf{v}}}^{\dagger})={{{\mathsf{L}}}}$. 3. 3.

Set dashboard allocation rule ${{{\mathnormal{y}}}}^{\dagger}$ as:

[TABLE]

Rebalancing dashboards constructed via Definition 14 satisfy the following properties.

Low-valued agents with value ${{\mathsf{v}}}\leq{{\mathsf{v}}}^{\dagger}$ bid nothing for ${{{\mathsf{L}}}}<0$ or bid their full value for ${{{\mathsf{L}}}}>0$. 2. 2.

High-valued agents with value ${{\mathsf{v}}}\geq{{\mathsf{v}}}^{\dagger}$ bid according to the rebalancing bid function ${{{\mathnormal{c}}}}^{\dagger}$ of Definition 9.

Our analysis of the outstanding balance for high-valued agents will be identical to the previous discussion of the reference rebalancing dashboard of Definition 9. Our analysis of the outstanding balance for low-valued agents is based on the following simple lemma which shows that either the balance is improved or the magnitude of the outstanding balance is at most the inferred value ${\hat{{\mathsf{v}}}}$, i.e., the same as the bound on the payment residual of Lemma 2.

Lemma 6**.**

With outstanding balance ${{{\mathsf{L}}}}$, dashboard ${{{\mathnormal{y}}}}$, allocation rule ${{{\mathnormal{x}}}}$, and inferred value ${\hat{{\mathsf{v}}}}\leq{{\mathsf{v}}}^{\dagger}$ from Definition 14; the magnitude of the subsequent outstanding balance of the rebalancing dashboard mechanism is either reduced or at most ${\hat{{\mathsf{v}}}}$.

Proof.

We prove the case of positive outstanding balance ${{{\mathsf{L}}}}>0$ (the negative balance case is analogous). We aim to show that the outstanding balance after this stage is between $-{\hat{{\mathsf{v}}}}$ and ${{{\mathsf{L}}}}$, i.e., the balance is reduced and if it becomes negative then its magnitude is at most ${\hat{{\mathsf{v}}}}$.

For $\gamma\to 1$ the bid according to dashboard ${{{\mathnormal{y}}}}^{\dagger}$ is ${{{\mathnormal{c}}}}^{\dagger}({\hat{{\mathsf{v}}}})\to{\hat{{\mathsf{v}}}}$. This is the most the agent could bid with value ${\hat{{\mathsf{v}}}}$ so ${{{\mathnormal{b}}}}({\hat{{\mathsf{v}}}})-{{{\mathnormal{c}}}}^{\dagger}({\hat{{\mathsf{v}}}})$ is non-positive. Thus, the balance is reduced.

It remains to show that if the outstanding balance becomes negative, i.e., we overshoot zero balance, that its magnitude is at most ${\hat{{\mathsf{v}}}}$. The largest value of ${\left|{{{\mathnormal{b}}}}({\hat{{\mathsf{v}}}})-{{{\mathnormal{c}}}}^{\dagger}({\hat{{\mathsf{v}}}})\right|}$ is ${\hat{{\mathsf{v}}}}$ which occurs when ${{{\mathnormal{b}}}}({\hat{{\mathsf{v}}}})=0$. Thus, the remaining outstanding balance is at least ${{{\mathsf{L}}}}-{\hat{{\mathsf{v}}}}$ which, since we started with a positive balance ${{{\mathsf{L}}}}>0$, is at least $-{\hat{{\mathsf{v}}}}$. ∎

To analyze the outstanding balance at stage $t$ to prove an analog to Theorem 4, we break the stages where the agent is allocated into three cases:

The agent has a low inferred value and after the stage the outstanding balance does not change sign. 2. 2.

The agent has a low inferred value and after the stage the outstanding balance changes sign. 3. 3.

The agent has a high inferred value.

Our analysis will be simple. In the first case, the magnitude of the outstanding balance does not increase. These stages can be ignored. Combining the stages where the other two cases hold, our previous analysis of Theorem 4 can be applied. To apply Theorem 4 we need the balance resolved to be within $[{{{\mathsf{L}}}}\,{{{\mathsf{\eta}}}},{{{\mathsf{L}}}}]$ and the magnitude of the payment residual, i.e., new balance that is generated, to be at most ${\hat{{\mathsf{v}}}}$. For stages that satisfy the second case we view the payments as exactly balancing ${{{\mathsf{L}}}}$ and then generating a new payment residual with magnitude at most ${\hat{{\mathsf{v}}}}$. For stages of the third case, the assumptions of the theorem are satisfied by the equivalence of the dashboard with the reference rebalancing dashboard (Definition 9) for the inferred value ${\hat{{\mathsf{v}}}}$. We conclude with the following theorem.

Theorem 5**.**

For any monotonic stage allocation rules ${{{\mathnormal{x}}}}^{(1)},\ldots,{{{\mathnormal{x}}}}^{(t)}$, any monotonic dashboard rules ${{{\mathnormal{y}}}}^{(1)},\ldots,{{{\mathnormal{y}}}}^{(t)}$ with probabilities supported on $[{{{\mathsf{\eta}}}},1]$, and any sequence of inferred values ${\hat{{\mathsf{v}}}}^{(1)},\ldots,{\hat{{\mathsf{v}}}}^{(t)}$ bounded by $\bar{\mathrm{{{\mathsf{v}}}}}$; the winner-pays-bid rebalancing dashboard has outstanding balance at stage $t$ of at most $\bar{\mathrm{{{\mathsf{v}}}}}/{{{\mathsf{\eta}}}}$; the rebalancing dashboard mechanism is $\bar{\mathrm{{{\mathsf{v}}}}}/({{{\mathsf{\eta}}}}t)$ incentive consistent.

6.5 Practical Considerations

In practical implementations of the rebalancing dashboard, it may be undesirable for the dashboard to fluctuate significantly from stage to stage as positive and negative balances are resolved. One approach to provide a dashboard that varies less across successive stages is to allow the magnitude outstanding balance to be a small positive multiple of ${\hat{{\mathsf{v}}}}$ and only resolve the outstanding balance when this allowed magnitude is exceeded. A detailed treatment of this approach is given in the next section.

7 Static Values: Natural Rebalancing

In this section we assume as in Section 5 and Section 6 that the functional form of the allocation algorithm is available. Like Section 5 the value of the agent of our analysis will be assumed to be static; like Section 6 the stage allocation rule will be assumed to be dynamic.

The rebalancing dashboard of Section 6 allowed the conversion of the dashboard corresponding to any strictly monotone allocation rule into one where the outstanding balance, i.e., the difference between payments in the sequential dashboard mechanism and sequential truthful mechanism, in any dynamic environment is bounded. On the other hand, following the dashboard for the inferred values dashboard of Section 5 converges in two stages to Nash and the subsequent payment residual is zero. In the static-value dynamic-allocation-rule environment, we identify dashboard with naturally low outstanding balance for an agent with static value that follows the dashboard. Thus, on the follow-the-dashboard equilibrium path, explicit rebalancing is only necessary when an agent’s value changes.

When combining the naturally rebalancing dashboards of this section with the rebalancing dashboards of Section 6, it is important not to attempt to resolve the payment residual immediately. For example, permitting a small outstanding balance of, e.g., $2\bar{\mathrm{{{\mathsf{v}}}}}$, to persist will allow the naturally rebalancing dashboard to balance itself. Of course, if the agent’s value changes, the payment residual can accumulate and will need to be resolved by explicit rebalancing.

7.1 All-pay Dashboards

The main result of this section is the observation that the all-pay last-stage inferred-values dashboard (Definition 4) does not require any rebalancing for an agent with a static value, even when the allocation algorithm or other agent values can be dynamic.

Lemma 7**.**

For an agent with static per-stage value ${{\mathsf{v}}}$, the outstanding balance of the all-pay last-stage inferred-values dashboard mechanism at stage $t$ is at most ${{\mathsf{v}}}$.

Proof.

We will consider grouping the allocation of stage $s$ with the payment in stage $s+1$ (with the allocation of stage $t$ paired with the payment of stage $1$) and we will refer to this grouping as “outcome $s$.” If the agent follows the dashboard, outcomes corresponding to $s\in\{1,\ldots,t-1\}$ are the truthful outcomes for the allocation rule of stage $s$ and value ${{\mathsf{v}}}$. Specifically, the bid in stage $s+1$ is equal to the payment according to the dashboard in stage $s+1$ which is equal to the allocation rule of stage $s$. Thus, the payment is stage $s+1$ is the correct payment for the allocation rule in stage $s$ and static value ${{\mathsf{v}}}$. The imbalance of payments from outcome $t$ (the allocation from stage $t$ and payment from stage 1), by Lemma 2 in Section 6, is at most ${{\mathsf{v}}}$. ∎

As is evident by the proof of Lemma 7, the payment residual in each stage can be large, but because of the relationship between the dashboard and allocation algorithm in successive stages the overall outstanding balance remains small.

7.2 Winner-pays-bid Dashboards

A similar approach admits a winner-pays-bid dashboard that does not require any rebalancing for an agent with a static value, even when the allocation algorithm or other agent values can be dynamic.

As a general principle, winner-pays-bid dashboard mechanisms tend work better when stages where an agent is not allocated are ignored. While, ignoring these stages when constructing the dashboard may make estimates of allocation rules less responsive, considering these stages in dynamic environments leaves open the possibility that the allocation rules for rounds where the agent is not allocated are systematically distinct from allocation rules in rounds where the ageht is allocated. This distinction can result in systematic differences between the dashboard and the allocation rule in stages where the agent is allocated and can result in large differences between the total payments in the sequential dashboard mechanism and the sequential truthful mechanism. The following dashboard is constructed based on only allocation rules where the agent wins.

Definition 15**.**

The last-winning-stage inferred-values dashboard in stage $t$ corresponds to the allocation rule from the latest stage $s<t$ where the agent was allocated by the mechanism.

Lemma 8**.**

For an agent with static per-stage value ${{\mathsf{v}}}$, the outstanding balance of the winner-pays-bid last-winning-stage inferred-values dashboard mechanism at stage $t$ is at most ${{\mathsf{v}}}$.

Proof.

Denote the stages that the agent is allocated, indexed in increasing order, by $\check{T}=\{s_{1},\ldots,s_{\check{\tau}}\}$. For convenience denote by $s_{0}=0$ and denote by ${{{\mathnormal{x}}}}^{(s_{0})}={{{\mathnormal{y}}}}^{(s_{1})}$ the initial dashboard. (This initial dashboard can be any arbitrary monotonic allocation rule.)

The total payments made by the agent are $\sum_{k=1}^{\check{\tau}}{{{\mathnormal{c}}}}^{(s_{k})}({{\mathsf{v}}})$. Here ${{{\mathnormal{c}}}}^{(s_{k})}$ is the bid strategy used in stage $s_{k}$ with the dashboard for ${{{\mathnormal{y}}}}^{(s_{k})}$. The bid strategy corresponding to the stage allocation rule ${{{\mathnormal{x}}}}^{(s_{k})}$ is ${{{\mathnormal{b}}}}^{(s_{k})}$; thus, the desired total payments are $\sum_{k=1}^{\check{\tau}}{{{\mathnormal{b}}}}^{(s_{k})}({{\mathsf{v}}})$.

For the last-winning-stage inferred-values dashboard ${{{\mathnormal{y}}}}^{(s_{k})}={{{\mathnormal{x}}}}^{(s_{k-1})}$; thus, ${{{\mathnormal{c}}}}^{(s_{k})}={{{\mathnormal{b}}}}^{(s_{k-1})}$ Thus, the difference in the total payments made and total payments desired is

[TABLE]

The magnitude of the difference between any two bids of an agent with value ${{\mathsf{v}}}$ is at most ${{\mathsf{v}}}$. The lemma follows. ∎

8 Single-call Dashboards

In this section we generalize the construction of dashboards and their analyses to the practically realistic case that the principal has only single-call access to the allocation algorithm. For example, given a profile of values, the principal can draw a single sample from the distribution of the algorithm’s outcomes. Such a dashboard is necessary for online marketplaces where the allocations of the algorithm are endogenous to the behavior of one side of the market. E.g., in ad auctions where the mechanism is designed for the advertisers and the users show up and make decisions that realize the stochasticity of the allocation. The single-call perspective of this section allows viewing the allocation algorithm as a map from the values of the agents to how agents are prioritized in the marketplace where stochastic outcomes are obtained.

The single-call model of mechanism design is one where the designer has an algorithm and aims to design a mechanism that implements the allocation of the algorithm in equilibrium. The designer, however, can only call the algorithm once. The single-call model is significant in that the standard methods for calculating payments in mechanisms, e.g., in the Vickrey-Clarke-Groves or unbiased payment mechanism (Archer and Tardos,, 2001), require $n+1$ blackbox calls to the algorithm. Babaioff et al., (2010) showed that truthful single-call mechanisms exist. These mechanisms randomly perturb the allocation algorithm and use these perturbations to compute the correct payments for the perturbed allocation algorithm.

There are two challenges to single-call implementation of dashboards. First, for the valuation profile input into the algorithm, only the realized allocation in $\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{x}}}}}\in\{0,1\}^{n}$ is observed. The allocation probabilities are not observed. Second, the principal does not have counterfactual access to the allocation algorithm that we have previously used to determine reasonable dashboards. Recall, a dashboard is a prediction of the bid allocation rule faced by a agent. This prediction requires knowledge, e.g., for agent $i$ of ${{{\mathcal{x}}}}_{i}(z,\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}_{-i})$ for all $z$.

Both of these challenges are solved by instrumenting the allocation algorithm. Specifically, we instrument the allocation algorithm with uniform exploration, i.e., with probably $\rho$ independently for each agent we enter a uniform random value rather than the agent’s value. This instrumentation degrades the quality of the allocation algorithm by $\rho$. First, this uniform instrumentation can be viewed as a randomized controlled experiment for estimating the counterfactual allocation rule as is necessary for giving the agent a dashboard that estimates her bid-allocation rule. Second, the uniform exploration enables implicitly calculating unbiased truthful payments for the realized mechanism (cf. Babaioff et al.,, 2010). The difference between these desired payments and the actual payments, i.e., the agent’s bid if she wins, can then be added to the outstanding balance in the payment rebalancing dashboard of Section 6.

Definition 16**.**

The instrumented allocation algorithm $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{w}}}}}$ for allocation algorithm $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}$, instrumentation rate $\rho\in(0,1)$, valuation range $[0,\bar{\mathrm{{{\mathsf{v}}}}}]$, and input valuation profile $\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}}$ is:

For each agent $i$, sample

[TABLE] 2. 2.

Define $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{w}}}}}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}})=\text{\bf E}_{\kern 0.3014pt{\check{{\kern-0.3014pt\bm{\mathsf{v}}}}}}\!\left[{\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}(\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}})}\right]$ and sample $\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{w}}}}}\sim\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{w}}}}}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}})$, i.e., run $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{x}}}}}$ on $\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}}$. 3. 3.

For each agent $i$, set instrumentation variables

[TABLE]

The truthful payment functions for $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{w}}}}}$ are denoted by $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathnormal{r}}}}}$, via the payment identity (2).

Theorem 6**.**

The instrumentation payment variables $\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{r}}}}}$ are unbiased estimators for the incentive compatible payments $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathnormal{r}}}}}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}})$ for the instrumented allocation algorithm $\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathcal{w}}}}}$, i.e., $\text{\bf E}\!\left[{\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{r}}}}}}\right]=\kern 0.43057pt{{{\kern-0.43057pt\bm{\mathnormal{r}}}}}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}})$.

Proof.

We break the payment ${{{\mathnormal{r}}}}_{i}(\kern 0.43057pt{{\kern-0.43057pt\bm{\mathsf{v}}}})$ for agent $i$ into two parts, the part where ${\check{{\mathsf{v}}}}_{i}={{\mathsf{v}}}_{i}$ and the part where ${\check{{\mathsf{v}}}}_{i}\neq{{\mathsf{v}}}_{i}$. In the latter part the payment identity requires zero payment. In the former, which happens with probability $1-\rho$, the payment conditioned on $\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i}$ is ${{{\mathcal{p}}}}_{i}({{\mathsf{v}}}_{i},\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i})={{\mathsf{v}}}_{i}\,{{{\mathcal{x}}}}_{i}({{\mathsf{v}}}_{i},\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i})-\int_{0}^{{{\mathsf{v}}}_{i}}{{{\mathcal{x}}}}_{i}(z,\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i})\,dz$. The expected payment conditioned on $\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i}$ (but not conditioning on ${\check{{\mathsf{v}}}}_{i}={{\mathsf{v}}}_{i}$) is $(1-\rho)\,{{{\mathcal{p}}}}_{i}({{\mathsf{v}}}_{i},\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i})$.

Now evaluate $\text{\bf E}\!\left[{{\check{{\mathsf{r}}}}_{i}\,\mid\,\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i}}\right]$ as defined in Definition 16 as follows:

[TABLE]

Since the expected payments conditioned on $\kern 0.43057pt{\check{{\kern-0.43057pt\bm{\mathsf{v}}}}}_{-i}$ are equal, so are the unconditional expected payments. ∎

Instrumentation allows the mechanism to construct a consistent estimator for the realized allocation rule that is otherwise unknown. This estimator can be used to construct a dashboard.

Definition 17**.**

For stage $t\geq t_{0}$, the instrumented dashboard is the profile of single-agent bid allocation rules $\kern 0.43057pt{\tilde{{\kern-0.43057pt\bm{\mathnormal{y}}}}}^{(t)}$ defined by ${\tilde{{\mathnormal{y}}}}_{i}^{(t)}$ for agent $i$ as follows:

Let ${\hat{{\mathsf{v}}}}_{i}^{(s)}$ denote the agent’s inferred value in stage $s\in\{1,\ldots,t-1\}$. 2. 1.

Define allocation data set corresponding to the uniform instrumentation $\{({\check{{\mathsf{v}}}}_{i}^{(s)},{\check{{\mathsf{w}}}}_{i}^{(s)}):s\in\{1,\ldots,t-1\}\wedge{\check{{\mathsf{v}}}}_{i}\neq{\hat{{\mathsf{v}}}}_{i}^{(s)}\}$. 3. 2.

Define the empirical average allocation as ${\hat{{\mathrm{\mu}}}}_{i}^{(t)}$ as the average allocation of this data set. 4. 3.

Define the empirical allocation rule as ${\hat{{\mathnormal{x}}}}_{i}^{(t)}:{\mathbb{R}}\to[0,1]$ as a continuous isotonic regression of the allocation data set. 5. 4.

Define the instrumented allocation rule as ${{{\mathnormal{y}}}}_{i}^{(t)}=(1-\rho)\,{\hat{{\mathnormal{x}}}}_{i}^{(t)}({{\mathsf{v}}})+\rho\,{\hat{{\mathrm{\mu}}}}_{i}^{(t)}$. 6. 5.

The instrumented dashboard for agent $i$ is the bid-allocation rule ${\tilde{{\mathnormal{y}}}}_{i}^{(t)}$ that corresponds to ${{{\mathnormal{y}}}}_{i}^{(t)}$ via equation (4).

Standard methods for isotonic regression can be used to estimate the empirical allocation rule in Definition 17. Errors in the regression will be resolved by the rebalancing approach of Section 6. A key required property, however, is that dashboard is continuous. Thus, isotonic regressions that result in continuous functions should be preferred. Approaches to isotonic regression in statistics, including smoothing and penalization (see Ramsay,, 1988, Mammen,, 1991, Kakade et al.,, 2011) guarantee continuity of the resulting estimator for the regression function. The ironing procedure common in Bayesian mechanism design is an isotonic regression; however, it results in discontinuous functions and is, thus, inappropriate for constructing dashboards.

The analysis below focuses on the winner-pays-bid rebalancing instrumented dashboard for the instrumented allocation algorithm. (Recall that all-pay mechanisms have deterministic payments and, thus, the budget imbalance generated at a given round can be resolved deterministically. We omit the simple analysis.) With winner-pays-bid mechanisms the payments are only made when the agent is allocated. As a result, the payment residual and balanced resolved are stochastic and necessitate a more sophisticated analysis.

The rebalancing approach in Section 6 used the functional form of the allocation algorithm to calculate the difference between actual payments given by the payment format and incentive compatible payments given by the payment identity. This payment residual is added to a balance and each stage a portion of the balance is added to the payment of a type with zero value when determining the dashboard. Our approach here is to instead use the implicit payments which, by Theorem 6, are unbiased estimators of the incentive compatible payments. The three terms in the balance update formula below are the previous balance, the implicit payment of the current stage, and the actual payment of the current stage.

Definition 18**.**

From stage $s$ with outstanding balance ${\check{{\mathsf{{{{\mathsf{L}}}}}}}}^{(s)}$, agent bid ${{\mathsf{b}}}^{(s)}$, realized instrumented allocation ${\check{{\mathsf{w}}}}^{(s)}$, and realized implicit payment ${\check{{\mathsf{r}}}}^{(s)}$; the stage $s+1$ outstanding balance in the winner-pays-bid instrumented rebalancing dashboard is:

[TABLE]

We first confirm that in expectation the analysis of Section 6 holds. Recall from equation (6) that the bid strategy for the payment rebalancing dashboard for allocation rule ${{{\mathnormal{y}}}}$ is ${{{\mathnormal{b}}}}^{\dagger}({{\mathsf{v}}})={{{\mathnormal{b}}}}({{\mathsf{v}}})+{\check{{\mathsf{{{{\mathsf{L}}}}}}}}\,{{{\mathsf{\eta}}}}/{{{\mathnormal{y}}}}({{\mathsf{v}}})$. Thus the actual payment in the balance update formula can be split for inferred value ${\hat{{\mathsf{v}}}}$ satisfying ${{\mathsf{b}}}={{{\mathnormal{b}}}}^{\dagger}({\hat{{\mathsf{v}}}})$ giving a balance update as

[TABLE]

Here ${{{\mathnormal{r}}}}$ is the payment rule for ${{{\mathnormal{w}}}}$ and payment ${{{\mathnormal{y}}}}({\hat{{\mathsf{v}}}})\,{{{\mathnormal{b}}}}({\hat{{\mathsf{v}}}})$ satisfies the payment identity for ${{{\mathnormal{y}}}}$. Thus, if the dashboard is correct, i.e., ${{{\mathnormal{y}}}}={{{\mathnormal{w}}}}$, then the expected payment residual $\text{\bf E}\!\left[{{\check{{\mathsf{d}}}}}\right]$ is zero. When we have incorrect estimates, the extent to which the difference of the first terms is not zero is gives a payment residual that must be rebalanced in the future.

Our analysis starts with three observations:

• •

When ${\check{{\mathsf{w}}}}=0$ the payment residual and amount rebalanced are zero, i.e., ${\check{{\mathsf{d}}}}={\check{{\mathsf{{{{\mathnormal{l}}}}}}}}=0$; otherwise:

• •

the size of the range of ${\check{{\mathsf{d}}}}$ is ${\hat{{\mathsf{v}}}}/\rho$ (equal to the size of the range of ${\check{{\mathsf{r}}}}$ which equals ${\hat{{\mathsf{v}}}}\,[-(1-\rho)/\rho,1]$ for inferred value ${\hat{{\mathsf{v}}}}$); and

• •

the amount rebalanced is ${\check{{\mathsf{{{{\mathsf{L}}}}}}}}{{{\mathsf{\eta}}}}/{{{\mathnormal{y}}}}({\hat{{\mathsf{v}}}})$ for inferred value ${\hat{{\mathsf{v}}}}$.

Our analysis proceeds like that of Section 6. We consider the $\check{\tau}$ stages $s$ where the agent is allocated, i.e., $\check{T}=\{s:{\check{{\mathsf{w}}}}^{(s)}\neq 0\}$. These are the stages where ${\check{{\mathsf{d}}}}^{(s)}$ and ${\check{{\mathsf{{{{\mathnormal{l}}}}}}}}^{(s)}$ can be non-zero. We consider the amount of the payment residual ${\check{{\mathsf{d}}}}^{(s)}$ from stage $s\in\check{T}$ that remains at final stage $t$ when, at each subsequent stage in $\check{T}$, a fraction of that payment residual is resolved. Lemma 4 shows that the balance resolved in each stage $s\in\check{T}$ is between ${{{\mathsf{L}}}}^{(s)}\,{{{\mathsf{\eta}}}}$ and ${{{\mathsf{L}}}}^{(s)}$. The magnitude of the payment residual is upper bounded by ${\hat{{\mathsf{v}}}}/\rho$ in the lemma below.

Lemma 9**.**

In a stage in which the agent’s inferred value is ${\hat{{\mathsf{v}}}}$, the single-call winner-pays-bid rebalancing dashboard for instrumented allocation algorithm with instrumentation probability $\rho$ has payment residual with magnitude $|{\check{{\mathsf{d}}}}|$ at most ${\hat{{\mathsf{v}}}}/\rho$.

Proof.

When ${\check{{\mathsf{x}}}}=0$ the payment residual is zero. The payment residual when ${\check{{\mathsf{x}}}}=1$ is ${\check{{\mathsf{d}}}}={\check{{\mathsf{r}}}}-{{{\mathnormal{b}}}}({\hat{{\mathsf{v}}}})$ with ${\check{{\mathsf{r}}}}={\hat{{\mathsf{v}}}}\,({\check{{\mathsf{x}}}}-\tfrac{1-\rho}{\rho}\,{\check{{\mathsf{\pi}}}})$. The magnitude $|{\check{{\mathsf{d}}}}|\leq\tfrac{1-\rho}{\rho}{\hat{{\mathsf{v}}}}+{{{\mathnormal{b}}}}({\hat{{\mathsf{v}}}})$. Since the winner-pays-bid strategy (without the addition for rebalancing) is individually rational ${{{\mathnormal{b}}}}({\hat{{\mathsf{v}}}})\leq{\hat{{\mathsf{v}}}}$ and, thus, $|{\check{{\mathsf{d}}}}|\leq{\hat{{\mathsf{v}}}}/\rho$. ∎

We are ready now to apply Theorem 4 to the rebalancing dashboard for the instrumented allocation algorithm. A helpful property of the instrumented allocation algorithm (Definition 16) is that the instrumentation implies that the allocation probabilities for all agents are bounded away from zero, i.e., the minimum allocation probability for an agent $i$ in the dashboard is $\rho$ times the average allocation probability for $i$ with value ${\check{{\mathsf{v}}}}_{i}\sim U[0,\bar{\mathrm{{{\mathsf{v}}}}}]$ in the induced allocation algorithm. We need to set ${{{\mathsf{\eta}}}}$ to lower bound this quantity.

Corollary 2**.**

For the single-call winner-pays-bid rebalancing dashboard for instrumented allocation algorithm with rebalancing rate ${{{\mathsf{\eta}}}}$ (set appropriately), instrumentation parameter $\rho$, and values in $[0,\bar{\mathrm{{{\mathsf{v}}}}}]$; the outstanding balance at stage $t$ is at most $\bar{\mathrm{{{\mathsf{v}}}}}/(\rho\,{{{\mathsf{\eta}}}})$.

Note that this bound is on the realized total balance and is not a high-probability or in-expectation result; it holds always. The payment residual at any stage is a random variable with expected magnitude at most $\bar{\mathrm{{{\mathsf{v}}}}}$ and range at most $\bar{\mathrm{{{\mathsf{v}}}}}/\rho$. The expected payment residual, however, has magnitude at most $\bar{\mathrm{{{\mathsf{v}}}}}$ (Lemma 2). Theorem 4 implies that the expected outstanding balance at time $t$ is at most $\bar{\mathrm{{{\mathsf{v}}}}}/{{{\mathsf{\eta}}}}$. Incentive consistency is defined in expectation over randomization in the mechanism and strategies. We have the following corollary. As before, holding $\bar{\mathrm{{{\mathsf{v}}}}}/{{{\mathsf{\eta}}}}$ constant, the incentive inconsistency vanishes with $t$.

Corollary 3**.**

For the single-call winner-pays-bid rebalancing dashboard for instrumented allocation algorithm with rebalancing rate ${{{\mathsf{\eta}}}}$ (set appropriately), instrumentation parameter $\rho$, and values in $[0,\bar{\mathrm{{{\mathsf{v}}}}}]$; over $t$ stages the dashboard mechanism is $\bar{\mathrm{{{\mathsf{v}}}}}/({{{\mathsf{\eta}}}}\,t)$ incentive inconsistent for all strategies.

The bound of Corollary 2 can be improved via the Chernoff-Hoeffding inequality. We show that the outstanding balance at time $t$ is the weighted average of these payment residuals with weights that are geometrically decreasing. This results in the following high-probability bound.

Theorem 7**.**

For the single-call winner-pays-bid rebalancing dashboard for instrumented allocation algorithm with rebalancing rate ${{{\mathsf{\eta}}}}$ (set appropriately), instrumentation parameter $\rho$, and values in $[0,\bar{\mathrm{{{\mathsf{v}}}}}]$; the outstanding balance at stage $t$ is at most $\bar{\mathrm{{{\mathsf{v}}}}}/{{{\mathsf{\eta}}}}+\frac{\bar{\mathrm{{{\mathsf{v}}}}}}{\rho}\sqrt{\frac{1}{2\eta}\log\left(\frac{2}{\delta}\right)}$ with probability at least $1-\delta$.

Proof.

By Lemma 9 for each stage $s$, $|{\check{{\mathsf{d}}}}^{(s)}|\leq\bar{\mathrm{{{\mathsf{v}}}}}/\rho.$ At the same time $\text{\bf E}\!\left[{{\check{{\mathsf{d}}}}^{(s)}}\right]\leq\bar{\mathrm{{{\mathsf{v}}}}}$ and ${\check{{\mathsf{d}}}}^{(s)}=0$ whenever ${\check{{\mathsf{w}}}}^{(s)}=0$. As in the proof of Theorem 4, consider the stages $\check{T}=\{s_{0},\ldots,s_{\check{\tau}-1}\}$ with non-zero payments indexed in decreasing order. Then

[TABLE]

where the range of each term in the sum is bounded by $(1-\eta)^{k}\,\bar{\mathrm{{{\mathsf{v}}}}}/\rho.$

Then we can apply the Chernoff-Hoeffding inequality to evaluate

[TABLE]

which follows from $\left(\sum\nolimits^{\check{\tau}}_{l=0}(1-{{{\mathsf{\eta}}}})^{2l}\right)^{-1}\geq\left(\sum\nolimits^{\infty}_{l=0}(1-{{{\mathsf{\eta}}}})^{2l}\right)^{-1}=1-(1-{{{\mathsf{\eta}}}})^{2}\geq{{{\mathsf{\eta}}}},$ since $0<{{{\mathsf{\eta}}}}<1$ and ${{{\mathsf{\eta}}}}\geq{{{\mathsf{\eta}}}}^{2}.$

Then with probability $\delta$ the imbalance of empirical dashboard with instrumentation can deviate from the expectation no further than $\frac{\bar{\mathrm{{{\mathsf{v}}}}}}{\rho}\sqrt{\frac{1}{2\eta}\log\left(\frac{2}{\delta}\right)}.$ ∎