# On Simultaneous Two-player Combinatorial Auctions

## TL;DR

This paper investigates the communication complexity of two-player combinatorial auctions, showing that simultaneous protocols can achieve the same approximation as interactive ones for welfare maximization, but decision problems are harder.

## Contribution

It demonstrates that simultaneous protocols can match the best interactive approximation guarantees, but decision problems require exponential communication, highlighting a separation in protocol power.

## Key findings

- Simultaneous protocols can achieve a 3/4-approximation for welfare maximization.
- Deciding welfare thresholds with high probability requires exponential communication in simultaneous protocols.
- There is a proven separation between the power of simultaneous and interactive protocols for these auction problems.

## Abstract

We consider the following communication problem: Alice and Bob each have some valuation functions $v_1(\cdot)$ and $v_2(\cdot)$ over subsets of $m$ items, and their goal is to partition the items into $S, \bar{S}$ in a way that maximizes the welfare, $v_1(S) + v_2(\bar{S})$. We study both the allocation problem, which asks for a welfare-maximizing partition and the decision problem, which asks whether or not there exists a partition guaranteeing certain welfare, for binary XOS valuations. For interactive protocols with $poly(m)$ communication, a tight 3/4-approximation is known for both [Fei06,DS06]. For interactive protocols, the allocation problem is provably harder than the decision problem: any solution to the allocation problem implies a solution to the decision problem with one additional round and $\log m$ additional bits of communication via a trivial reduction. Surprisingly, the allocation problem is provably easier for simultaneous protocols. Specifically, we show: 1) There exists a simultaneous, randomized protocol with polynomial communication that selects a partition whose expected welfare is at least $3/4$ of the optimum. This matches the guarantee of the best interactive, randomized protocol with polynomial communication. 2) For all $\varepsilon > 0$, any simultaneous, randomized protocol that decides whether the welfare of the optimal partition is $\geq 1$ or $\leq 3/4 - 1/108+\varepsilon$ correctly with probability $> 1/2 + 1/ poly(m)$ requires exponential communication. This provides a separation between the attainable approximation guarantees via interactive ($3/4$) versus simultaneous ($\leq 3/4-1/108$) protocols with polynomial communication. In other words, this trivial reduction from decision to allocation problems provably requires the extra round of communication.