Uninformed Player Research Articles

Knowing the informed player's payoffs and simple play in repeated games

We revisit the canonical model of repeated games with two patient players, observable actions, and one-sided incomplete information, and make the substantive assumption that the informed player's preference is state independent. We show the informed player can attain a payoff in equilibrium if and only if she can attain it in the simple class of equilibria first studied by Aumann, Maschler, and Stearns (1968), in which the initial stages are used only for revealing information, and no further information is revealed after the initial stages. This sufficiency result does not extend to the uninformed player's equilibrium payoff set.

Bargaining with Voluntary Disclosure and Endogenous Matching

We investigate a bargaining setting between an “informed” player, who has private information, and an “uninformed” player. The informed player has the option to truthfully disclose private information in two unique environments. In the first environment, the informed player is randomly matched with an uninformed player and, after matching, can voluntarily disclose private information prior to a negotiation taking place. In the second environment, the informed player can voluntarily disclose private information before any endogenous matching between players (and subsequent negotiation) takes place. We begin by examining these environments theoretically. When disclosure occurs after matching, we show that low informed types should disclose more often than high informed types to avoid disagreement. However, when disclosure occurs before matching, for a range of parameter values, we show that high informed types should disclose more than low informed types to secure a better match. We test these predictions in a controlled human-subjects experiment and verify many of the theoretical predictions. Among other results, we find that low informed types do indeed disclose their private information to avoid disagreement and that, when it is in their interest to do so, high informed types disclose their private information to secure a better match. Another key insight is that high uninformed types benefit from disclosure regardless of whether it occurs before or after matching. This paper was accepted by Axel Ockenfels, behavioral economics and decision analysis. Supplemental Material: The online appendix and data files are available at https://doi.org/10.1287/mnsc.2022.03566 .

Testing rational hypotheses in signaling games

We introduce a solution concept for signaling games, called Rational Hypothesis Testing Equilibrium (RHTE). Beliefs are updated via Ortoleva’s (2012) Hypothesis Testing model, allowing for conditioning on information sets off the path. Hypotheses are conjectures by the uninformed player about opponent’s strategies that rationalize sending an unexpected message. Each RHTE is a Perfect Bayesian Equilibrium, but not vice versa. RHTE features a number of desirable properties: First, beliefs are structurally consistent in the spirit of Kreps and Wilson (1982). Second, beliefs are consistent with mutual knowledge of rationality. Third, RHTE can be related to the prominent refinement concepts, including the Intuitive Criterion, strategic stability, and undefeated equilibrium. In the Spence game, RHTE restricts the admissible wages, significantly reducing the number of equilibria. Finally, we show that our equilibrium notion offers an alternative explanation for the experimental results in Brandts and Holt (1992, 1993).

Two-person zero-sum risk-sensitive stochastic games with incomplete reward information on one side

This study considers zero-sum risk-sensitive discrete-time stochastic games with incomplete reward information on one side. We show the existence of the value function, derive a new Shapley equation, and prove that the value function solves the Shapley equation, which is used to construct an optimal policy for the informed player. For the uninformed player, we construct an optimal policy by introducing a dual game. Finally, we give an example to illustrate the effects of the reward information and risk-sensitive parameters on the value function and optimal policies.

A Signaling Theory of the Online Consumer Review Policy

Abstract In this paper, we consider a two-period model of an experience good with a seller (informed player) and a consumer (uninformed player) in each period. In the model, we examine the seller’s decision about offering refunds for online reviews of his products, and each period consumer’s purchasing decision together with the first period consumer’s reviewing decision. Our main interest is whether a high-quality product seller offers a high amount or a low amount of cashback for an online review. We show that a lenient cashback policy for a review can be a signal of high quality of the product. Intuitively, a high-quality seller can offer a higher amount of cashback to a consumer who reviews, whether the review is positive or negative. This separation is possible mainly due to a difference in the second-period profits across types. We also briefly discuss the effect of the conditional review policy fostering fake reviews.

Information spillover in multiple zero-sum games

This paper considers an infinitely repeated three-player zero-sum game with two-sided incomplete information, in which an informed player plays two zero-sum games simultaneously at each stage against two uninformed players. This is a generalization of the model in Aumann et al. (Repeated games with incomplete information. MIT Press, New York, 1995) of two-player zero-sum one-sided incomplete information games. Under a correlated prior, the informed player faces the problem of how to optimally disclose information among two uninformed players in order to maximize his long-term average payoffs (i.e., undiscounted payoffs). Our objective is to understand the adverse effects of “information spillover” from one game to the other in the equilibrium payoff set of the informed player. We provide conditions under which the informed player can fully overcome such adverse effects and characterize equilibrium payoffs. In a second result, we show how the effects of information spillover on the equilibrium payoff set of the informed player might be severe. Finally, we compare our findings on the equilibrium-payoff set of the informed player with those of Bayesian Persuasion models with multiple receivers.

Bargaining with Mechanisms

Two players bargain over a single indivisible good and a transfer, with one-sided incomplete information about preferences. Both players can offer arbitrary mechanisms to determine the allocation. We show that there is a unique perfect Bayesian equilibrium outcome. In the equilibrium, one of the players proposes a menu that is optimal for the uninformed player among all menus, such that each type of the informed player receives at least her payoff under complete information. The optimal menu can be implemented with at most three allocations. Under a natural assumption on the uninformed player’s beliefs, the optimal menu coincides with the Myerson’s neutral solution to the bargaining problem in this environment. (JEL C78, D82, D83)

Information disclosure in sequential (and simultaneous) contests

We study a two-player contest with one-sided incomplete information: One player’s valuation is publicly known and the other’s is his private information. The designer chooses the players’ move sequence (between simultaneous and sequential contests) and the information disclosure policy (between full and no disclosure). We find that if the uninformed player’s valuation is low, a simultaneous (resp. strong-lead sequential) contest with full disclosure is optimal when the valuation difference is large (resp. small); if the uninformed player’s valuation is high, a strong-lead sequential contest with full (resp. no) disclosure is optimal when the valuation difference is large (resp. small).

Dynkin Games with Incomplete and Asymmetric Information

We study the value and the optimal strategies for a two-player zero-sum optimal stopping game with incomplete and asymmetric information. In our Bayesian setup, the drift of the underlying diffusion process is unknown to one player (incomplete information feature), but known to the other one (asymmetric information feature). We formulate the problem and reduce it to a fully Markovian setup where the uninformed player optimises over stopping times and the informed one uses randomised stopping times in order to hide their informational advantage. Then we provide a general verification result that allows us to find the value of the game and players’ optimal strategies by solving suitable quasi-variational inequalities with some nonstandard constraints. Finally, we study an example with linear payoffs, in which an explicit solution of the corresponding quasi-variational inequalities can be obtained.

Knowing the Informed Player's Payoffs and Simply Play in Repeated Games

We revisit the classic model of two-player repeated games with undiscounted utility, observable actions, and one-sided incomplete information, and further assume the informed player has state-independent preferences. We show the informed player can attain a payoff in equilibrium if and only if she can attain it in the simple class of equilibria first studied by Aumann, Maschler, and Stearns (1968), in which information is only revealed in the game's initial stages. This sufficiency result does not extend to the uninformed player's equilibrium payoff set.

Reputation effects in stochastic games with two long-lived players

This paper extends the reputation result of Evans and Thomas (Econometrica 65(5):1153–1173, 1997) to stochastic games. Specifically, I analyze reputation effects with two long-lived (but not equally long) players in stochastic games, that is, in games in which the payoffs depend on a state variable and the law of motion for states is a function of the current state and of the players’ actions. The results suggest that reputation effects in stochastic games can be expected if the uninformed player has a weak control of the law motion or if there is no irreversibility. On the contrary, in economic situations with irreversibility and in which the uninformed player’s strong control of the transition, such as the hold-up problems, reputation effects are unlikely to be obtained.

Resilience of LTE eNode B against smart jammer in infinite-horizon asymmetric repeated zero-sum game

It has been proposed to use Long Term Evolution (LTE)/LTE-Advanced (LTE-A) networks for mission critical and public safety applications. However, LTE/LTE-A networks are known to be vulnerable to denial-of-service (DOS) and loss-of-service attacks from smart jammers. This article deals with the resilience of LTE/LTE-A eNode B against smart jamming attacks in an infinite-horizon asymmetric repeated zero-sum game and introduces algorithms for constructing efficient strategies for both players (smart jammer and eNode B) in such a game. It has been shown in game-theoretic literature that security strategies provide optimal solution in zero-sum games and that both players’ security strategies in an infinite-horizon asymmetric repeated zero-sum game depend only on the history of informed player’s actions. However, fixed-sized sufficient statistics are needed for both players to solve an infinite-horizon game efficiently with memory constraints. Smart jammer (informed player) uses its evolving belief state as the fixed-sized sufficient statistic for the repeated game. Whereas, LTE eNode B (uninformed player) uses worst-case regret of its security strategy and its anti-discounted update as the fixed-sized sufficient statistic. Although fixed-sized sufficient statistics are exploited by both players, optimal security strategy computation in λ-discounted asymmetric games is still hard to compute because of non-convexity. Hence, the problem is convexified by devising suboptimal security strategies with guaranteed performance for both players that are based on approximated optimal game value. However, “approximated” strategies require full monitoring. Therefore, a simplistic yet effective “expected” strategy is also constructed for LTE eNode B (uninformed player) that does not require full monitoring. The simulation results show that smart jammer maintains its dominance at a long range of prior probability values by playing non-revealing and misleading strategies against the network for its long-term advantage.

Reputation Effects Under Interdependent Values

A patient player privately observes a persistent state and interacts with an infinite sequence of myopic uninformed players. The patient player is either a strategic type who maximizes his payoff or one of several commitment types who mechanically play the same action in every period. I focus on situations in which the uninformed player's best reply to a commitment action depends on the state and where the total probability of commitment types is sufficiently small. I show that the patient player's equilibrium payoff is bounded below his commitment payoff in some equilibria under some of his payoff functions. This is because he faces a trade‐off between building his reputation for commitment and signaling favorable information about the state. When players' stage‐game payoff functions are monotone‐supermodular, the patient player receives high payoffs in all states and in all equilibria. Under an additional condition on the state distribution, my reputation model yields a unique prediction on the patient player's equilibrium payoff and on‐path behavior.

Optimal Stopping and Strategic Espionage

We study a continuous-time reputation game in which an informed player bene ts from persistent private information while an uninformed player suffers from it. Observing noisy signals of the informed player's actions, the uninformed player chooses when to reveal the informed player's private information, but revelation is costly. We characterize the sequential equilibrium in Markov strategies, provide comparative statistics of equilibrium, and study the probability that the uninformed player reveals the informed player's private information. In addition, we contrast our model with Kyletype models in terms of informativeness of signals and analyze the extended model in which observation is costly.

Efficient Strategy Computation in Zero-Sum Asymmetric Information Repeated Games

Zero-sum asymmetric information games model decision-making scenarios involving two competing players who have different information about the game being played. A particular case is that of nested information, where one (informed) player has superior information over the other (uninformed) player. This paper considers the case of nested information in repeated zero-sum games and studies the computation of strategies for both the informed and uninformed players for finite-horizon and discounted infinite-horizon nested information games. For finite-horizon settings, we exploit that for both players, the security strategy, and also the opponent's corresponding best response, depend only on the informed player's history of actions. Using this property, we formulate an linear program (LP) computation of player strategies that is linear in the size of the uninformed player's action set. For the infinite-horizon discounted game, we construct LP formulations to compute the approximated security strategies for both players, and show that the worst-case performance difference between the approximated security strategies and the security strategies converges to zero exponentially. Finally, we illustrate the results on a network interdiction game between an informed system administrator and an uniformed intruder.

Information Spillover in a Bayesian Repeated Game

This paper considers an infinitely repeated three-player Bayesian game with lack of information on two sides, in which an informed player plays two zero-sum games simultaneously at each stage against two uninformed players. This is a generalization of Aumann, Maschler and Stearns (1995) two-player zero-sum one-sided incomplete information model. Under a correlated prior, the informed player faces the problem of how to optimally disclose information among two uninformed players in order to maximize his long term average payoffs. The objective is to understand the adverse effects of “information spillover” from one game to the other in the equilibrium payoff set of the informed player. We provide conditions under which the informed player can fully overcome such adverse effects, and show that in some cases the adverse effects are unsurmountable and severe.

Zero-sum games with ambiguity

We study whether the basic tenets of the theory of zero-sum games are still valid when players are uncertainty averse. We focus on games with one-sided information, in which the uninformed player is uncertain about the state of nature. Uncertainty aversion turns the underlying zero-sum game into a many-player, non-zero-sum game. We show that the uninformed player has a unique equilibrium payoff. We provide conditions under which there is a unique equilibrium payoff vector.

Information Transmission in Revision Games

Revision games model a situation in which players can prepare their actions during a pre-play phase. We introduce one-sided incomplete information in two coordination games, one of common interest and one of opposing interest, and study how the pre-play phase affects coordination. We find that in the common interest game, the unique Bayesian equilibrium is such that the informed player will signal the state of the world through her prepared action, unless the pre-play phase is about to finish, in which case she seeks to coordinate with the other player. In the opposing interest game, the equilibrium is similar when the informed player is the one receiving less opportunity to revise her actions. When it is the uninformed player who receives less revision opportunities, we show that it is possible no information is revealed if both players are initially coordinated, but some information must be revealed if they are initially miscoordinated.

When and how much the altruism impacts your privileged information? Proposing a new paradigm in game theory: The “boxers game”

In this work, we proposed a new N-person game in which the players can bet on two options, for example represented by two boxers. Some of the players have privileged information about the boxers and part of them can provide this information to uninformed players. However, this information may be true if the informed player is altruist or false if he is selfish. So, in this game, the players are divided in three categories: informed and altruist players, informed and selfish players, and uninformed players. By considering the matchings (N∕2 distinct pairs of randomly chosen players) and that the payoff of the winning group follows aspects captured from two important games, the public goods game and minority game, we showed quantitatively and qualitatively how the altruism can impact on the privileged information. We localized analytically the regions of positive payoffs which were corroborated by numerical simulations performed for all values of information and altruism densities given that we know the information level of the informed players. Finally, in an evolutionary version of the game ,we showed that the gain of the informed players can get worse if we adopted the following procedure: the players increase their investment for situations of positive payoffs, and decrease their investment when negative payoffs occur.

Jammer-Type Estimation in LTE With a Smart Jammer Repeated Game

Long-term evolution (LTE)/LTE-Advanced (LTE-A) networks are known to be vulnerable to denial-of-service (DOS) and loss-of-service attacks from smart jammers. The interaction between the network and the smart jammer has been modeled as an infinite-horizon general-sum (nonzero-sum) Bayesian game with asymmetric information, with the network being the uninformed player. Although significant work has been done on optimal strategy computation and control of information revelation of the informed player in repeated asymmetric information games, it has been limited to zero-sum games with perfect monitoring. Recent progress on the strategy computation of the uninformed player is also limited to zero-sum games with perfect monitoring and is focused on expected payoff formulations. Since the proposed formulation is a general-sum game with imperfect monitoring, existing formulations cannot be leveraged for estimating true state of nature (the jammer type). Hence, a threat-based mechanism is proposed for the uninformed player (the network) to estimate the informed player's type (jammer type). The proposed mechanism helps the network resolve uncertainty about the state of nature (jammer type) so that it can compute a repeated-game strategy conditioned on its estimate. The proposed algorithm does not rely on the commonly assumed “full monitoring” premise and uses a combination of threat-based mechanism and nonparametric estimation to estimate the jammer type. In addition, it requires no explicit feedback from the network users, nor does it rely on a specific distribution (e.g., Gaussian) of test statistic. It is shown that the proposed algorithm's estimation performance is quite robust under realistic modeling and observational constraints, despite all the aforementioned challenges.