Contrarian Motives in Social Learning

TL;DR

This paper analyzes how individuals with contrarian motives influence social learning, showing that such motives can lead to increased information acquisition and reduced conformity, affecting decision thresholds and behaviors.

Contribution

It introduces a model of social learning incorporating nonconformity motives, revealing how contrarian incentives alter information acquisition and action choices.

Findings

Contrarian motives can increase the value of information and signal investment.

Stronger contrarian motives decrease the likelihood of conforming to the current majority.

Optimal actions shift linearly with observed popularity and contrarian intensity.

Abstract

We study sequential social learning with endogenous information acquisition when agents have a taste for nonconformity. Each agent observes predecessors' actions, chooses whether to acquire a private signal (and its precision), and then selects between two actions. Payoffs reward correctness and add a history-based bonus for taking the less popular action, so equilibrium inference remains Bayesian without fixed points in anticipated popularity. In a Gaussian-quadratic specification, optimal actions are posterior thresholds that shift linearly with observed popularity and contrarian intensity, tilting decisions against the majority. We solve the precision choice problem with a fixed entry cost and a convex cost of precision. Whenever the no-signal action coincides with the observed majority, stronger contrarian motives weakly increase the maximized value of information and enlarge the set of histories in which agents invest in signals. We also derive comparative statics for thresholds and choice probabilities. In particular, increasing contrarian intensity reduces the likelihood of taking the currently popular action in both states.

Equations

$$s_t\mid \theta \sim \mathcal{N}(\theta ,1/{\rho }_t),$$

$$p_t(a)=\{\begin{array}{ll}\frac{N_{t-1}(a)}{t-1},& t\ge 2,\\ \frac{1}{2},& t=1,\end{array}$$

$$b(p_t(a_t))=k(1-p_t(a_t)),k\ge 0.$$

$$u_t(a_t,\theta ;h_{t-1})=1\{a_t=\theta \}+k(1-p_t(a_t)).$$

$${\mu }_t=\mathrm{Pr}(\theta =1\mid h_{t-1};\sigma ).$$

$$\mathrm{Pr}(\theta =1\mid h_{t-1},s_t;\sigma )=\Lambda \left(\log \frac{{\mu }_t}{1-{\mu }_t}+{\ell }_t(s_t;{\rho }_t)\right),$$

$$({\phi }_t,{\alpha }_t):{\mathcal{H}}_{t-1}\to {\mathbb{R}}_{+}\times \mathcal{A},$$

$${\Delta }_t:=b(p_t(1))-b(p_t(0))=k(1-p_t)-k{p}_t=k(1-2p_t).$$

$$\underset{\text{correctness term}}{\underbrace{2\mathrm{Pr}(\theta =1\mid h_{t-1},s)-1}}+\underset{\text{contrarian bonus difference}}{\underbrace{{\Delta }_t}}.$$

$$\mathrm{Pr}(\theta =1\mid h_{t-1},s)\ge c_t,c_t=\frac{1-{\Delta }_t}{2}=\frac{1}{2}+k(p_t-\frac{1}{2}).$$

$$L_t+{\ell }_t(s;\rho )\ge {\tau }_t,{\tau }_t:=\log \frac{c_t}{1-c_t}.$$

$$s_t^{\star }(\rho )=\frac{1}{2}+\frac{{\tau }_t-L_t}{\rho },$$

$$G_t({\mu }_t,\rho ;p_t,k):=\underset{\text{correctness}}{\underbrace{{\mu }_t\mathrm{Pr}(a_t=1\mid \theta =1)+(1-{\mu }_t)\mathrm{Pr}(a_t=0\mid \theta =0)}}+k\underset{\text{contrarian bonus}}{\underbrace{\mathbb{E}\left[1-p_t(a_t)\right]}},$$

$$\underset{\rho \ge 0}{\max }\{G_t({\mu }_t,\rho ;p_t,k)-\frac{c}{2}{\rho }^2-F1\{\rho >0\}\}.$$

$$s_t^{\star }(\rho )=\frac{1}{2}+\frac{{\tau }_t-L_t}{\rho },z_0(\rho ):=s_t^{\star }(\rho )\sqrt{\rho },z_1(\rho ):=(s_t^{\star }(\rho )-1)\sqrt{\rho },$$

$$\mathrm{Pr}(a_t=1\mid \theta =1)$$

$$\mathrm{Pr}(a_t=1\mid \theta =0)$$

$$\mathrm{Pr}(a_t=0\mid \theta =0)$$

$$\mathrm{Pr}(a_t=1)={\mu }_t[1-\Phi (z_1(\rho ))]+(1-{\mu }_t)[1-\Phi (z_0(\rho ))].$$

$$\mathbb{E}[1-p_t(a_t)]=(1-p_t)\mathrm{Pr}(a_t=1)+p_t(1-\mathrm{Pr}(a_t=1))=p_t+(1-2p_t)\mathrm{Pr}(a_t=1).$$

$$\frac{\partial }{\partial \rho }{G}_t({\mu }_t,\rho ;p_t,k){|}_{\rho ={\rho }_t^{\star }}=c{\rho }_t^{\star },$$

$$\frac{\partial }{\partial \rho }{G}_t({\mu }_t,\rho ;p_t,k)$$

$$+k(1-2p_t)\underset{\frac{\partial }{\partial \rho }\mathrm{Pr}(a_t=1)}{\underbrace{\left[{\mu }_t\varphi \right(z_1(\rho ))(-z_1^{\prime }(\rho ))+(1-{\mu }_t)\varphi (z_0(\rho ))(-z_0^{\prime }(\rho ))]}}.$$

$$z_0^{\prime }(\rho )=\frac{s_t^{\star }(\rho )}{2\sqrt{\rho }}+\sqrt{\rho }{s}_t^{\star \prime }(\rho ),z_1^{\prime }(\rho )=\frac{s_t^{\star }(\rho )-1}{2\sqrt{\rho }}+\sqrt{\rho }{s}_t^{\star \prime }(\rho ),s_t^{\star \prime }(\rho )=-\frac{{\tau }_t-L_t}{{\rho }^2}.$$

$${\Phi }_t({\mu }_t,p_t;k):=\underset{\rho \ge 0}{\max }\{G_t({\mu }_t,\rho ;p_t,k)-\frac{c}{2}{\rho }^2\}-G_t({\mu }_t,0;p_t,k).$$

$$\begin{array}{rl}{\Phi }_t({\mu }_t,p_t;k)& =\underset{\rho \ge 0}{\sup }\{A_t(\rho )+k{B}_t(\rho )-\frac{c}{2}{\rho }^2\},\\ A_t(\rho ):& ={\text{Corr}}_t({\mu }_t,\rho )-{\text{Corr}}_t({\mu }_t,0),\\ B_t(\rho ):& =M_t({\mu }_t,\rho ;p_t)-M_t({\mu }_t,0;p_t).\end{array}$$

$$\frac{\partial c_t}{\partial k}=p_t-\frac{1}{2},\frac{\partial {\tau }_t}{\partial k}=\frac{p_t-\frac{1}{2}}{c_t(1-c_t)},\frac{\partial s_t^{\star }}{\partial k}=\frac{p_t-\frac{1}{2}}{\rho c_t(1-c_t)}.$$

$$\frac{\partial }{\partial k}{P}_1^1(\rho )=-\varphi (z_1(\rho ))\frac{1}{\sqrt{\rho }}\frac{p_t-\frac{1}{2}}{c_t(1-c_t)},\frac{\partial }{\partial k}{P}_1^0(\rho )=-\varphi (z_0(\rho ))\frac{1}{\sqrt{\rho }}\frac{p_t-\frac{1}{2}}{c_t(1-c_t)}.$$

$$s_t^{\star }(\rho )=\frac{1}{2}+\frac{{\tau }_t-L_t}{\rho },z_0(\rho )=s_t^{\star }(\rho )\sqrt{\rho },z_1(\rho )=(s_t^{\star }(\rho )-1)\sqrt{\rho }.$$

$$\mathrm{Pr}(a_t=1\mid \theta =1)=1-\Phi (z_1(\rho )),\mathrm{Pr}(a_t=0\mid \theta =0)=\Phi (z_0(\rho )),$$

$$\frac{\partial }{\partial \rho }{G}_t({\mu }_t,\rho ;p_t,k)={\mu }_t\phi (z_1)(-z_1^{\prime })+(1-{\mu }_t)\phi (z_0)z_0^{\prime }+k(1-2p_t)\frac{\partial }{\partial \rho }\mathrm{Pr}(a_t=1),$$

$$\mathbb{E}[1-p_t(a_t)]-\mathbb{E}[1-p_t(a_t)]{|}_{\rho =0}=(1-2p_t)(P_1(\rho )-1)\ge 0.$$

Full text

Keywords: social learning; information cascades; endogenous information acquisition; nonconformity; popularity; Bayesian thresholds.

JEL codes: D83; C72; D82; D85.

1 Introduction

Sequential social learning shows how rational agents can end up herding on the basis of observed actions. Once the information contained in past actions dominates what a typical private signal would add, later actions often stop reacting to new information. Yet in many settings agents have a taste for doing something different from the crowd: scientists may seek distinctiveness, consumers may value uniqueness, and political actors may position themselves against a majority. This paper studies how such contrarian motives interact with social learning when information acquisition is itself a choice.

We develop a simple, fully specified model in which each agent arrives sequentially, decides whether to buy a private signal and how precise it should be, and then chooses a binary action. Agents value correctness and also derive a bonus from taking the less popular action, where popularity is the fraction of predecessors who chose a given action. The bonus is proportion-based, increasing in the size of the opposing crowd, and depends only on observed past actions. This design avoids any fixed-point in “anticipated popularity” and keeps Bayesian inference disciplined by the actual history. We work in a Gaussian–quadratic specification for tractability and transparency.

The analysis proceeds in two steps. First, we characterize action choice. Optimal behavior takes the form of a posterior (equivalently, log-likelihood ratio) threshold that depends on the observed popularity of the action (Lemma 4.1). The indifference cutoff shifts linearly with popularity and with the intensity of contrarian preferences, tilting decisions away from the observed majority (Proposition 4.2). We also quantify how the associated signal threshold moves with these parameters and how the induced action probabilities respond (Corollary 5.1 and Corollary 5.2).

Second, we study information acquisition. We formulate the precision choice problem with a fixed entry cost and a convex cost of precision and establish existence of an optimal precision together with an interior first-order condition in the Gaussian case (Proposition 4.3). We then define the net value of information at a given history and show that, whenever the no-signal action coincides with the observed majority, this value is (weakly) increasing in the strength of contrarian motives; hence the investment region expands with contrarian intensity in that empirically relevant region (Proposition 4.4). Taken together, these results deliver a clear positive message. Contrarian motives shift decision thresholds against the crowd and thereby sustain experimentation precisely in those histories where, absent contrarianism, agents would be more prone to follow predecessors. At the same time, by making majority actions harder to choose, stronger contrarian motives can reduce the incidence of majority choices even when the observed crowd is informative.

2 Related literature

This paper connects three strands of work: social learning from actions, endogenous information acquisition, and preference-based (non)conformity.

The baseline we build on is the sequential social-learning tradition in which actions reveal private signals and can trigger herding and cascades. Foundational contributions include Banerjee, (1992) and Bikhchandani et al., (1992), with sharper characterizations in Smith and Sørensen, (2000, 2011). Extensions examine learning with partial observability and networks, e.g. Acemoglu et al., (2011) and Dasaratha and He, (2020). A complementary literature studies why learning fails under misspecification or coarse inference even without payoff externalities, including Bohren, (2016), Frick et al., (2020), Eyster et al., (2014), Bohren et al., (2019), and Kartik and Van Weelden, (2020). Our contribution keeps Bayesian inference fully disciplined by observed actions and asks how a taste for nonconformity shifts the thresholds that govern action choice and, in turn, the informativeness of actions and the incentives to gather information.

We adopt a standard “fixed + convex” information technology to endogenize whether agents buy signals and how precise these signals are. Classic work on timing and informational externalities in sequential environments includes Chamley and Gale, (1994). For formal treatments of information choice and precision, see Vives, (2008) and Veldkamp, (2011); recent empirical evidence on belief-dependent information acquisition in markets is provided by Lu et al., (2019). Within this tradition, our novelty is to show how observable, taste-based contrarian motives tilt decision thresholds and thereby expand the region of beliefs at which information acquisition is worthwhile, while preserving closed-form characterizations of action thresholds and regularity of the precision problem.

Classic conformity arises when actions signal type or status, as in Bernheim, (1994), or when identity considerations affect payoffs, as in Akerlof and Kranton, (2000) and Bénabou and Tirole, (2011). We invert the sign and study contrarian preferences within a sequential learning game. Related models of distinctiveness and anti-conformism outside the canonical cascade environment include Golman et al., (2017) and Touboul, (2019). A separate set of papers blends social learning with technological payoff externalities (coordination/complementarities), e.g. Dasgupta et al., (2011) and Ali and Kartik, (2012). Our approach is different:111In coordination/complementarity settings (e.g., Ali and Kartik, (2012)), an agent’s payoff depends on others’ actions. Here, preferences add a taste term that depends only on the observed popularity of one’s own chosen action, leaving Bayesian inference and the history-to-belief mapping unchanged. we add a taste-based bonus tied purely to observed popularity among predecessors, so the inference problem and equilibrium mapping from histories to beliefs remain standard, and there is no fixed-point in anticipated popularity. This modeling choice isolates how minority-seeking shifts cutoffs and affects information incentives without conflating preference shifts with payoff externalities.

Finally, it is useful to clarify when motives are contrarian rather than conformist. In applications where distinctiveness itself carries rents or identity value—scientific priority races, niche political positioning, or consumption domains emphasizing uniqueness—the minority action is privately valuable even holding informational incentives fixed. Our specification captures exactly this case by rewarding the observed minority choice, while leaving the Bayesian learning channel unchanged.

The paper is organized as follows. Section 3 presents the environment. Section 4 characterizes equilibrium behavior and information acquisition. Section 5 records comparative statics. Section 6 concludes. The appendix collects proofs.

3 Model

We study a canonical sequential social-learning environment in which agents endogenously choose whether to acquire information (and how precise it is) before taking a binary action. Agents value correctness and, in addition, have a taste for nonconformity that depends on the observed popularity of the action they choose among predecessors.

3.1 Environment and information

Time is discrete, with agents $t=1,2,\ldots$ arriving sequentially. The state $\theta\in\{0,1\}$ is fixed and drawn once at $t=1$. The common prior is $\mu_{1}\equiv\Pr(\theta=1)=\tfrac{1}{2}$ (all results extend to any $\mu_{1}\in(0,1)$ with only notational changes). Each agent observes the entire history of past actions $h_{t-1}=(a_{1},\dots,a_{t-1})$, where $a_{i}\in\{0,1\}$.

Before acting, agent $t$ may acquire a private signal of endogenous precision $\rho_{t}\geq 0$. If $\rho_{t}=0$, no signal is acquired. If $\rho_{t}>0$, the signal is Gaussian:222Normality is chosen for closed-form thresholds and clean comparative statics. All results that rely on single-crossing/MLRP extend to any signal family with MLRP; formulas then become implicit but the qualitative monotone effects are unchanged.

[TABLE]

conditionally independent across agents given $\theta$. Let $\ell_{t}(s_{t};\rho_{t})$ denote the log-likelihood ratio (LLR), which in this Gaussian specification equals $\ell_{t}(s_{t};\rho_{t})=\rho_{t}\,(s_{t}-\tfrac{1}{2})$ (equal variances and means [math] and $1$).

Acquiring any signal entails a fixed cost $F>0$. Precision is chosen at a convex cost $C(\rho)=\tfrac{c}{2}\rho^{2}$ with $c>0$.333Any strictly convex $C$ with $C(0)=C^{\prime}(0)=0$ yields the same comparative statics; quadratic cost is a transparent benchmark. The fixed cost $F>0$ generates the empirically relevant corner at $\rho=0$; setting $F=0$ collapses to a pure convex case with the same Euler condition. Thus the total information cost is $F\,\mathbf{1}\{\rho>0\}+\tfrac{c}{2}\rho^{2}$.

3.2 Preferences and contrarian utility

An agent’s payoff has two components:

• •

Correctness: the agent obtains $1$ if $a_{t}=\theta$ and [math] otherwise.

• •

Nonconformity: let $N_{t-1}(a)=\sum_{i=1}^{t-1}\mathbf{1}\{a_{i}=a\}$ and define the observed popularity of action $a$ among predecessors at time $t$ by

[TABLE]

so that $p_{t}(0)+p_{t}(1)=1$ for $t\geq 2$ and both actions are treated symmetrically at $t=1$. The contrarian bonus is proportional to how unpopular the chosen action is:444Linear $b(p)=k(1-p)$ is a convenient normalization. Any smooth, strictly decreasing $b$ with $b^{\prime}(p)<0$ delivers the same first-order threshold tilt around interior beliefs; linearity makes the dependence on $p_{t}$ and $k$ fully transparent.

[TABLE]

Hence agent $t$’s (ex post) utility is

[TABLE]

The bonus depends only on observed predecessors’ actions. It does not depend on contemporaneous or future actions, so there is no fixed-point in defining the bonus.555If the bonus depended on expected contemporaneous popularity, equilibrium would embed a fixed-point in popularity beliefs and could generate multiplicity unrelated to learning. Tying $b(\cdot)$ to observed predecessors isolates the informational channel.

3.3 Timing

At date $t$:

The public history $h_{t-1}$ is observed. The public belief is $\mu_{t}=\Pr(\theta=1\mid h_{t-1})$ (defined below). 2. 2.

Agent $t$ chooses a precision $\rho_{t}\geq 0$, pays $F\,\mathbf{1}\{\rho_{t}>0\}+\frac{c}{2}\rho_{t}^{2}$, and if $\rho_{t}>0$ observes $s_{t}\sim\mathcal{N}(\theta,1/\rho_{t})$. 3. 3.

Agent $t$ chooses $a_{t}\in\{0,1\}$ and receives $u_{t}(a_{t},\theta;h_{t-1})$. 4. 4.

$a_{t}$ is publicly revealed; the history updates to $h_{t}=(h_{t-1},a_{t})$.

3.4 Beliefs and posteriors

Let $\sigma$ denote a strategy profile (defined below). Given $\sigma$, the public belief after history $h_{t-1}$ is

[TABLE]

If agent $t$ acquires precision $\rho_{t}$ and observes $s_{t}$, her posterior is

[TABLE]

where $\Lambda(x)=\frac{e^{x}}{1+e^{x}}$ is the logistic function and $\ell_{t}(s_{t};\rho_{t})=\rho_{t}\,(s_{t}-\tfrac{1}{2})$ is the Gaussian LLR. If $\rho_{t}=0$ (no signal) the posterior equals the public belief, i.e., $\Pr(\theta=1\mid h_{t-1};\sigma)=\mu_{t}$.

3.5 Strategies and equilibrium

A (pure) strategy for agent $t$ is a pair of measurable maps

[TABLE]

where $\mathcal{H}_{t-1}$ is the set of feasible histories and $\mathcal{A}$ is the set of Borel-measurable action rules $\alpha_{t}:\mathbb{R}\cup\{\varnothing\}\to\{0,1\}$ mapping either a realized signal $s_{t}\in\mathbb{R}$ (if $\rho_{t}=\varphi_{t}(h_{t-1})>0$) or the null input $\varnothing$ (if $\rho_{t}=0$) into an action. We write $\rho_{t}=\varphi_{t}(h_{t-1})$.

An assessment $(\sigma,\mu)$ consists of a profile of strategies $\sigma=\{\varphi_{t},\alpha_{t}\}_{t\geq 1}$ and a system of public beliefs $\{\mu_{t}\}_{t\geq 1}$.

Equilibrium concept.

An assessment $(\sigma,\mu)$ is a Perfect Bayesian Equilibrium666All statements carry over to sequential equilibrium. We restrict to pure strategies throughout; randomization matters only at indifference points. (PBE) if: (i) For each $t$ and history $h_{t-1}$, $\big(\varphi_{t}(h_{t-1}),\alpha_{t}(\cdot\mid h_{t-1})\big)$ maximizes agent $t$’s expected utility net of information costs, given $(\sigma,\mu)$; (ii) $\mu$ is obtained from Bayes’ rule whenever applicable, given $\sigma$ and the model’s likelihoods; off-path beliefs are specified in a way consistent with standard refinements. We adopt the tie-breaking convention that an indifferent agent picks the action with the higher contrarian bonus (the less popular action);777Any fixed tie-breaking rule yields the same results generically; it only bites on knife-edge histories (measure zero under continuous signals). results are unchanged under any fixed tie-breaking rule.

3.6 Popularity statistic and basic definitions

Let $p_{t}(a)$ be as defined above. We use $p_{t}\equiv p_{t}(1)$ as a shorthand for the popularity of action $1$ at time $t$, so $p_{t}(0)=1-p_{t}$. The popularity statistic is fully observable at time $t$.

Definition 3.1**.**

At history $h_{t-1}$, agent $t$’s action is informative if, conditional on the public belief $\mu_{t}$ and the strategy profile $\sigma$, there exist signal realizations leading to each action with strictly positive probability; equivalently, both $\Pr(a_{t}=1\mid\theta=1,h_{t-1};\sigma)$ and $\Pr(a_{t}=1\mid\theta=0,h_{t-1};\sigma)$ lie strictly between [math] and $1$.

Definition 3.2**.**

A $1$-cascade occurs at history $h_{t-1}$ if $a_{t}=1$ almost surely under $\sigma$ for both states; a [math]-cascade is defined analogously. In a cascade, actions are uninformative.888With endogenous information, contrarian motives can disrupt would-be cascades by restoring incentives to acquire signals. Proposition 4.4 formalizes the region where this happens.

3.7 Standing assumptions

We maintain the following assumptions throughout the paper:

1. (A1)

Signals. Conditional on $\theta$, private signals are conditionally independent across agents; if precision $\rho>0$ is chosen, $s\mid\theta\sim\mathcal{N}(\theta,1/\rho)$. 2. (A2)

Costs. $F>0$ and $c>0$; the cost of precision is $C(\rho)=\tfrac{c}{2}\rho^{2}$. 3. (A3)

Preferences. The contrarian bonus is $b(p)=k(1-p)$ with $k\geq 0$, applied to the observed popularity of the chosen action among predecessors. 4. (A4)

Observability and common knowledge. The entire action history $h_{t-1}$ is public; signals are private; $(F,c,k)$ and the model are common knowledge.

These assumptions, combined with the Gaussian specification, imply that posterior beliefs are monotone in $s_{t}$ (MLRP)999With asymmetric priors $\mu_{1}\neq\tfrac{1}{2}$ all formulas go through with the log-odds shift $L_{t}=\log\frac{\mu_{t}}{1-\mu_{t}}$; we normalize $\mu_{1}=\tfrac{1}{2}$ for symmetry in the exposition. and that optimal action rules can be represented as posterior- or LLR-thresholds in equilibrium. Formal statements and proofs appear in Section 4.

4 Equilibrium analysis

We characterize optimal action choice as a posterior/LLR threshold, derive the dependence of the threshold on the observed popularity $p_{t}$, and analyze the information acquisition problem (existence, first-order condition, and comparative statics in $k$ for the value of information under clear baseline conditions).

Throughout the section fix a date $t$ and a public history $h_{t-1}$ with associated public belief $\mu_{t}\in(0,1)$, log-odds $L_{t}:=\log\frac{\mu_{t}}{1-\mu_{t}}$, and observed popularity $p_{t}(1)=N_{t-1}(1)/(t-1)$ (with the convention $p_{1}(1)=1/2$). Write $p_{t}:=p_{t}(1)$ and $p_{t}(0)=1-p_{t}$.

4.1 Action choice: posterior and LLR thresholds

At history $h_{t-1}$, the bonus differential from choosing $1$ rather than [math] is

[TABLE]

Let $c_{t}$ be the posterior cutoff that makes the agent indifferent between actions.

Lemma 4.1**.**

Fix a history $h_{t-1}$. For any chosen precision $\rho\geq 0$ and any realized signal $s$, the difference in expected utility between actions $1$ and [math] equals

[TABLE]

Hence the optimal action is a posterior threshold: choose $a_{t}=1$ iff

[TABLE]

Equivalently, in LLR units choose $a_{t}=1$ iff

[TABLE]

In the Gaussian case with $\ell_{t}(s;\rho)=\rho\,(s-\tfrac{1}{2})$, this is the signal threshold rule $a_{t}=1$ iff $s\geq s_{t}^{\star}(\rho)$ where

[TABLE]

with the convention that $s_{t}^{\star}(0)=+\infty$ if $\mu_{t}<c_{t},\;-\infty$ if $\mu_{t}>c_{t}$.

Proof.

Let $\pi(s):=\Pr(\theta=1\mid h_{t-1},s)$ denote the posterior. The expected correctness payoff of $a=1$ is $\pi(s)$ and of $a=0$ is $1-\pi(s)$, so the difference is $2\pi(s)-1$. The expected contrarian bonus of $a=1$ is $k(1-p_{t})$ and of $a=0$ is $k(1-p_{t}(0))=k\,p_{t}$, so the bonus difference is $\Delta_{t}=k(1-2p_{t})$, which does not depend on $s$ or $\rho$. Therefore the agent prefers $1$ to [math] iff $2\pi(s)-1+\Delta_{t}\geq 0$, i.e., $\pi(s)\geq(1-\Delta_{t})/2=:c_{t}$. Since posteriors in a binary-state, MLRP environment are strictly increasing in the LLR, the posterior cutoff implies the LLR cutoff $L_{t}+\ell_{t}(s;\rho)\geq\tau_{t}:=\log\frac{c_{t}}{1-c_{t}}$. In the Gaussian case $\ell_{t}(s;\rho)=\rho(s-\tfrac{1}{2})$ gives the claimed $s$-threshold. ∎

Proposition 4.2**.**

For any history, $c_{t}=\frac{1}{2}+k(p_{t}-\frac{1}{2})$. Hence: (i) $c_{t}$ is strictly increasing in $p_{t}$ when $k>0$; (ii) if $p_{t}>\frac{1}{2}$ then $c_{t}>\frac{1}{2}$ and $c_{t}$ is strictly increasing in $k$; if $p_{t}<\frac{1}{2}$ then $c_{t}<\frac{1}{2}$ and $c_{t}$ is strictly decreasing in $k$; (iii) if $p_{t}=\frac{1}{2}$, then $c_{t}=\frac{1}{2}$ for all $k$.

Proof.

Immediate from Lemma 4.1 since $\Delta_{t}=k(1-2p_{t})$. ∎

4.2 Information acquisition: existence, FOC, and probabilities

Given $(\mu_{t},p_{t})$ and $\rho\geq 0$, define the gross expected payoff

[TABLE]

where probabilities are induced by the Gaussian signal and the threshold rule of Lemma 4.1. The agent then solves

[TABLE]

We first make the Gaussian probabilities explicit. Let

[TABLE]

and let $\Phi$ and $\phi$ denote the standard normal cdf and pdf, respectively. Then

[TABLE]

and the unconditional choice probability is

[TABLE]

Hence the expected contrarian bonus equals

[TABLE]

Proposition 4.3**.**

For any $(\mu_{t},p_{t},k)$, the information acquisition problem admits a solution $\rho_{t}^{\star}\geq 0$. Moreover, if $\rho_{t}^{\star}>0$ solves the problem, then it satisfies the first-order condition

[TABLE]

where

[TABLE]

Here $z_{0}^{\prime}(\rho)$ and $z_{1}^{\prime}(\rho)$ are given by

[TABLE]

Proof sketch.

Continuity and the quadratic cost imply existence of a maximizer. In the Gaussian case, write the state-conditional choice probabilities using the signal threshold $s_{t}^{\star}(\rho)$. Differentiating $G_{t}$ yields the Euler condition $\partial_{\rho}G_{t}=c\rho$ at any interior optimum. Full derivatives appear in Appendix A. ∎

4.3 Value of information and its monotonicity in $k$

Define the (net) value of information at $(\mu_{t},p_{t},k)$ by

[TABLE]

Observe that $G_{t}(\mu_{t},\rho;p_{t},k)=\text{Corr}_{t}(\mu_{t},\rho)\;+\;k\,M_{t}(\mu_{t},\rho;p_{t})$, where $M_{t}(\mu_{t},\rho;p_{t}):=\mathbb{E}[1-p_{t}(a_{t})]$. Hence

[TABLE]

The sign of $B_{t}(\rho)$ depends on whether the no-signal action aligns with the observed majority or minority. The following proposition isolates a robust region where $\Phi_{t}$ is (weakly) increasing in $k$ and therefore the investment region expands with $k$.

Proposition 4.4**.**

Fix $(\mu_{t},p_{t})$ with $p_{t}\neq\tfrac{1}{2}$ and let $a_{t}^{0}$ be the action chosen when $\rho=0$ (i.e., $a_{t}^{0}=1$ iff $\mu_{t}\geq c_{t}$). If $a_{t}^{0}$ coincides with the observed majority action at $h_{t-1}$ (i.e., $a_{t}^{0}=1$ when $p_{t}>\tfrac{1}{2}$, or $a_{t}^{0}=0$ when $p_{t}<\tfrac{1}{2}$), then for all $\rho\geq 0$, $B_{t}(\rho)\geq 0$ and thus $k\mapsto\Phi_{t}(\mu_{t},p_{t};k)$ is (weakly) increasing. Consequently, for any fixed $F>0$, the set $\{\;k\geq 0:\ \Phi_{t}(\mu_{t},p_{t};k)>F\;\}$ is an interval of the form $[k_{0},\infty)$ (possibly $k_{0}=0$ or $+\infty$).

Proof sketch.

When the baseline action follows the observed majority, raising $\rho$ weakly reduces the expected bonus shortfall at $\rho=0$, so $G_{t}$ has increasing differences in $(\rho,k)$. Taking the supremum over $\rho\geq 0$ preserves monotonicity in $k$. See Appendix A for details. ∎

Remark 4.5*.*

If the no-signal action coincides with the minority (e.g., $p_{t}>\tfrac{1}{2}$ and $\mu_{t}<c_{t}$, so $a_{t}^{0}=0$), then $P_{1}(0)=0$ and the same algebra yields $B_{t}(\rho)\leq 0$ for all $\rho\geq 0$. In that case $k\mapsto\Phi_{t}(\mu_{t},p_{t};k)$ need not be increasing. Proposition 4.4 cleanly separates the region where monotonic expansion in $k$ is guaranteed from the complementary region where no general monotonicity can be claimed.

5 Comparative statics

We collect a few immediate implications for thresholds, signal cutoffs, and the signal cutoffs and choice probabilities.

Corollary 5.1**.**

For any history with popularity $p_{t}\in[0,1]$ and $k\geq 0$:

[TABLE]

Hence, if $p_{t}>\tfrac{1}{2}$ (action $1$ is the observed majority), then $c_{t}$ and $\tau_{t}$ increase in $k$ and the signal threshold $s_{t}^{\star}$ shifts up in $k$ (it becomes harder to choose the majority action $1$). If $p_{t}<\tfrac{1}{2}$, the inequalities reverse. When $p_{t}=\tfrac{1}{2}$, all three derivatives are zero.

Proof.

From Lemma 4.1, $c_{t}=\tfrac{1}{2}+k(p_{t}-\tfrac{1}{2})$ so $\partial c_{t}/\partial k=p_{t}-\tfrac{1}{2}$. Since $\tau_{t}=\log\frac{c_{t}}{1-c_{t}}$, the chain rule gives $\partial\tau_{t}/\partial k=(\partial\tau_{t}/\partial c_{t})(\partial c_{t}/\partial k)=\frac{1}{c_{t}(1-c_{t})}(p_{t}-\tfrac{1}{2})$. Finally, $s_{t}^{\star}(\rho)=\tfrac{1}{2}+(\tau_{t}-L_{t})/\rho$ implies $\partial s_{t}^{\star}/\partial k=(1/\rho)\,\partial\tau_{t}/\partial k$. ∎

Corollary 5.2**.**

Fix $(\mu_{t},p_{t},k,\rho)$ and let $P_{1}^{\theta}(\rho):=\Pr(a_{t}=1\mid\theta)$ under the Gaussian threshold. Then

[TABLE]

Thus if $p_{t}>\tfrac{1}{2}$ (majority is action $1$), increasing $k$ reduces the likelihood of choosing $1$ in both states, and vice versa when $p_{t}<\tfrac{1}{2}$.

Proof.

$P_{1}^{1}(\rho)=1-\Phi(z_{1}(\rho))$ and $P_{1}^{0}(\rho)=1-\Phi(z_{0}(\rho))$ with $z_{\iota}(\rho)=\big(s_{t}^{\star}(\rho)-\iota\big)\sqrt{\rho}$, $\iota\in\{0,1\}$. By the chain rule, $\partial_{k}[1-\Phi(z_{\iota})]=-\phi(z_{\iota})\,\partial_{k}z_{\iota}$ and $\partial_{k}z_{\iota}=\sqrt{\rho}\,\partial_{k}s_{t}^{\star}$. Substitute $\partial_{k}s_{t}^{\star}$ from Corollary 5.1. ∎

6 Conclusion

We study sequential social learning with endogenous information acquisition when agents have a taste for nonconformity that depends on the observed popularity of actions. Because the bonus is purely history-based, Bayesian updating and the mapping from histories to public beliefs remain standard.

In a Gaussian–quadratic specification, optimal actions follow posterior (equivalently, log-likelihood ratio) thresholds that tilt against the observed majority: higher contrarian intensity raises the cutoff for taking the currently popular action and lowers it for taking the unpopular one. On the information margin, we characterize the precision choice under a fixed entry cost and a convex cost of precision, and obtain an interior first-order condition in the Gaussian case. Moreover, whenever the no-signal action coincides with the observed majority, the (maximized) value of information is weakly increasing in contrarian intensity, so the set of histories in which agents invest in private signals expands with contrarian motives in that empirically relevant region.

Several extensions are natural. Allowing heterogeneity (and possibly private information) in contrarian intensity, introducing networks of observation, or adding forward-looking incentives could help connect the framework to richer empirical settings.

Appendix A Additional proofs for Section 4

A.1 Proof of Proposition 4.3

The objective $G_{t}(\mu_{t},\rho;p_{t},k)-\tfrac{c}{2}\rho^{2}-F\mathbf{1}\{\rho>0\}$ is continuous on $[0,\infty)$ and goes to $-\infty$ as $\rho\to\infty$, so a maximizer exists. In the Gaussian case define

[TABLE]

Then

[TABLE]

and $\mathbb{E}[1-p_{t}(a_{t})]=p_{t}+(1-2p_{t})\Pr(a_{t}=1)$. Differentiating gives

[TABLE]

with $z_{0}^{\prime}=\frac{s_{t}^{\star}}{2\sqrt{\rho}}+\sqrt{\rho}\,s_{t}^{\star\prime}$, $z_{1}^{\prime}=\frac{s_{t}^{\star}-1}{2\sqrt{\rho}}+\sqrt{\rho}\,s_{t}^{\star\prime}$, and $s_{t}^{\star\prime}=-(\tau_{t}-L_{t})/\rho^{2}$. At any interior maximizer $\rho_{t}^{\star}>0$ the Euler condition $\partial_{\rho}G_{t}=c\rho_{t}^{\star}$ holds.

A.2 Proof of Proposition 4.4

Let $P_{1}(\rho)=\Pr(a_{t}=1)$. If $p_{t}>\tfrac{1}{2}$ and the no-signal action is $1$, then $P_{1}(0)=1$ and

[TABLE]

Thus the objective inside the $\sup_{\rho\geq 0}$ has increasing differences in $(\rho,k)$, and the supremum $\Phi_{t}(\mu_{t},p_{t};k)$ is weakly increasing in $k$ (the case $p_{t}<\tfrac{1}{2}$ is symmetric). Berge’s maximum theorem yields continuity and the interval property of the investment region.