Matching Pennies Game Research Articles

How hosts and pathogens choose the strengths of defense and counterdefense: a game-theoretical view

Host–pathogen interactions consist of an attack by the pathogen, frequently a defense by the host and possibly a counterdefense by the pathogen. Here, we present a game-theoretical approach to describe such interactions. We consider a game where the host and pathogen are players and can choose between the strategies of defense (or counterdefense) and no response. Specifically, they may or may not produce a toxin and an enzyme degrading the toxin, respectively. We consider that the host and pathogen must also incur a cost for toxin or enzyme production. We highlight both the sequential and non-sequential versions of the game and determine the Nash equilibria. Furthermore, we resolve a paradox occurring in that interplay. If the inactivating enzyme is very efficient, producing the toxin becomes useless, leading to the enzyme being no longer required. Then, the production of the defense becomes useful again. In game theory, such situations can be described by a generalized matching pennies game. As a novel result, we find under which conditions the defense cycle leads to a steady state or an oscillation. We obtain, for saturating dose–response kinetics and considering monotonic cost functions, “partial (counter)defense” strategies as pure Nash equilibria. This implies that producing a moderate amount of toxin and enzyme is the stable situation in this game.

Non-parametric identification and testing of quantal response equilibrium

This paper studies the falsifiability and identification of Quantal Response Equilibrium (QRE) when each player's utility and error distribution are relaxed to be unknown non-parametric functions. Using the variation of players' choices across a series of games, we first show that both the utility function and the distribution of errors are non-parametrically over-identified. This over-identification result further suggests a straightforward testing procedure for QRE which achieves the desired type-1 error and maintains a small type-2 error. To apply this methodology, we conduct an experimental study of the matching pennies game. Our non-parametric estimates strongly reject the conventional Logit choice probability. Moreover, when the utility and the error distribution are sufficiently flexible and heterogeneous, the quantal response hypothesis cannot be rejected for 70% of participants. However, strong assumptions such as linear utility, logistically distributed errors, and homogeneity lead to substantially higher rejection rates.

Rational Inattention in the Infield

This paper provides evidence of rational inattention by experienced professionals in strategic interactions. We add rational inattention to a game of matching pennies with state-dependent payoffs. Unlike the full-information, mixed-strategy Nash equilibrium, payoffs of different actions need not be equated state by state. Moreover, players respond partially to payoff differences, this responsiveness is stronger when attention costs are lower, strategies converge to full-information Nash as stakes increase, and average payoffs across all states are approximately equal across actions. We test these predictions using data on millions of pitches from Major League Baseball, where we observe strategies, payoffs, and proxies for attention costs. (JEL C72, D83, D91, L83, Z21)

Probability matching and strategic decision making

This paper examines a link between an individual’s (possibly limited) strategic thinking in the 11–20 money request game and (possibly non-rational) decision-making patterns in the matching pennies games. Experimental evidence shows that subjects’ strategic behavior, which used to be understood as a result of finite cognitive iterations, is closely related to their choice randomization patterns. Ignoring some individuals’ choice randomization may bias the population variance of levels in cognitive iterations. Choice randomizers, which we call probability matchers, are non-rational in both the non-strategic and strategic settings, but their choice patterns are systematic—although inconsistent with rational decisions—and similar.

Nash equilibria in human sensorimotor interactions explained by Q-learning with intrinsic costs

The Nash equilibrium concept has previously been shown to be an important tool to understand human sensorimotor interactions, where different actors vie for minimizing their respective effort while engaging in a multi-agent motor task. However, it is not clear how such equilibria are reached. Here, we compare different reinforcement learning models to human behavior engaged in sensorimotor interactions with haptic feedback based on three classic games, including the prisoner’s dilemma, and the symmetric and asymmetric matching pennies games. We find that a discrete analysis that reduces the continuous sensorimotor interaction to binary choices as in classical matrix games does not allow to distinguish between the different learning algorithms, but that a more detailed continuous analysis with continuous formulations of the learning algorithms and the game-theoretic solutions affords different predictions. In particular, we find that Q-learning with intrinsic costs that disfavor deviations from average behavior explains the observed data best, even though all learning algorithms equally converge to admissible Nash equilibrium solutions. We therefore conclude that it is important to study different learning algorithms for understanding sensorimotor interactions, as such behavior cannot be inferred from a game-theoretic analysis alone, that simply focuses on the Nash equilibrium concept, as different learning algorithms impose preferences on the set of possible equilibrium solutions due to the inherent learning dynamics.

Fairness under uncertainty

With the help of a repeated matching pennies game, we model a situation in which one of two players may accidentally get into an adverse situation that the other player can take advantage of. In doing so, we vary the degree of uncertainty about future rounds and especially about the frequency and role distribution of situations in which fairness and positive reciprocity are possible. It turns out that uncertainty can greatly increase the number of fair moves when unfair behavior leads to a loss of efficiency. If there is no loss of efficiency and if unfairness only leads to a redistribution between players, we observe almost exclusively selfish behavior. We also vary the social distance and find that small distance leads to a significant increase in the proportion of fair moves even in games where there is no loss of efficiency.

Believing in Bits: Digital Media and the Supernatural

Believing in Bits: Digital Media and the Supernatural

Endogenous quantal response equilibrium

We endogenize the precision parameter λ of logit quantal response equilibrium (LQRE). In the first stage of an endogenous quantal response equilibrium (EQRE), each player chooses her precision optimally subject to costs, taking as given other players’ (second-stage) behavior. In the second stage, the distribution of players’ actions is a heterogenous LQRE given the profile of first-stage precision choices. EQRE satisfies a modified version of the regularity axioms, nests LQRE as a limiting case for a sequence of cost functions, and admits analogues of classic results for LQRE such as those for equilibrium selection. We show how EQRE differs from LQRE using the family of generalized matching pennies games.

Identification of Non-Equilibrium Beliefs in Games of Incomplete Information Using Experimental Data

Abstract This paper studies the identification of players’ preferences and beliefs in discrete choice games using experimental data. The experiment comprises a set of games that differ in their matrices of monetary payoffs. The researcher is interested in the identification of preferences (utility of money) and beliefs on the opponents’ expected behavior, without imposing equilibrium restrictions or parametric assumptions on utility and belief functions. We show that the hypothesis of unbiased/rational beliefs is testable as long as the set of games in the experiment imply variation in monetary payoffs of other players, keeping the own monetary payoff constant. We present conditions for the full identification of utility and belief functions at the individual level – without restrictions on players’ heterogeneity in preferences or beliefs. We apply our method to data from two experiments: a matching pennies game, and a public good game.

Behavioural and neural interactions between objective and subjective performance in a Matching Pennies game

To examine the behavioural and neural interactions between objective and subjective performance during competitive decision-making, participants completed a Matching Pennies game where win-rates were fixed within three conditions (win > lose, win = lose, win < lose) and outcomes were predicted at each trial. Using random behaviour as the hallmark of optimal performance, we observed item (heads), contingency (win-stay, lose-shift) and combinatorial (HH, HT, TH, TT) biases across all conditions. Higher-quality behaviour represented by a reduction in combinatorial bias was observed during high win-rate exposure. In contrast, over-optimism biases were observed only in conditions where win rates were equal to, or less than, loss rates. At a group level, a neural measure of outcome evaluation (feedback-related negativity; FRN) indexed the binary distinction between positive and negative outcome. At an individual level, increased belief in successful performance accentuated FRN amplitude differences between wins and losses. Taken together, the data suggest that objective experiences of, or, subjective beliefs in, the predominance of positive outcomes may be mutual attempts to self-regulate performance during competition. In this way, increased exposure to positive outcomes (real or imagined) may help to weight the output of the more diligent and analytic System 2, relative to the impulsive and intuitive System 1.

Balancing model-based and memory-free action selection under competitive pressure.

In competitive situations, winning depends on selecting actions that surprise the opponent. Such unpredictable action can be generated based on representations of the opponent's strategy and choice history (model-based counter-prediction) or by choosing actions in a memory-free, stochastic manner. Across five different experiments using a variant of a matching-pennies game with simulated and human opponents we found that people toggle between these two strategies, using model-based selection when recent wins signal the appropriateness of the current model, but reverting to stochastic selection following losses. Also, after wins, feedback-related, mid-frontal EEG activity reflected information about the opponent's global and local strategy, and predicted upcoming choices. After losses, this activity was nearly absent-indicating that the internal model is suppressed after negative feedback. We suggest that the mixed-strategy approach allows negotiating two conflicting goals: 1) exploiting the opponent's deviations from randomness while 2) remaining unpredictable for the opponent.

How Much Would You Pay to Change a Game before Playing It?

Envelope theorems provide a differential framework for determining how much a rational decision maker (DM) is willing to pay to alter the parameters of a strategic scenario. We generalize this framework to the case of a boundedly rational DM and arbitrary solution concepts. We focus on comparing and contrasting the case where DM’s decision to pay to change the parameters is observed by all other players against the case where DM’s decision is private information. We decompose DM’s willingness to pay a given amount into a sum of three factors: (1) the direct effect a parameter change would have on DM’s payoffs in the future strategic scenario, holding strategies of all players constant; (2) the effect due to DM changing its strategy as they react to a change in the game parameters, with the strategies of the other players in that scenario held constant; and (3) the effect there would be due to other players reacting to a the change in the game parameters (could they observe them), with the strategy of DM held constant. We illustrate these results with the quantal response equilibrium and the matching pennies game and discuss how the willingness to pay captures DM’s anticipation of their future irrationality.

Games, graphs and Kirchhoff laws

Evolutionary potential games represent a set of biological and ecological models equivalent to multiparticle physical systems for a suitable dynamical rule. In these systems the pair interaction is described by a payoff matrix of two-player games possessing a wider class of interactions. Potential games satisfy criteria related to the Kirchhoff laws and have pure Nash equilibria. Using the bi-matrix formalism of game theory we show a simple method for checking the existence of potential which is related to the absence of cyclic components. It will be shown that potential exists if the game is orthogonal to a suitable set of cycling elementary games resembling voluntary matching pennies games. Relationships among these cyclic components and consequences of player’s equivalence are also discussed.

TESTING THE QUANTAL RESPONSE HYPOTHESIS

AbstractWe develop a nonparametric test for consistency of player behavior with the quantal response equilibrium (QRE). The test exploits a characterization of the equilibrium choice probabilities in any structural QRE as the gradient of a convex function; thereby, QRE‐consistent choices satisfy the cyclic monotonicity inequalities. Our testing procedure utilizes recent econometric results for moment inequality models. We assess our test using lab experimental data from a series of generalized matching pennies games. We reject the QRE hypothesis in the pooled data but cannot reject individual‐level quantal response behavior for over half of the subjects.

A Collective Quantum Matching Pennies Game

This article studies an iterative collective matching pennies (MP) quantum game, where every player interacts with his neighbours. The MP game evolves by allowing every player to adopt the strategy of his best paid mate. The behaviour in fair and unfair contests is examined.

Topology-dependent rationality and quantal response equilibria in structured populations.

Given that the assumption of perfect rationality is rarely met in the real world, we explore a graded notion of rationality in socioecological systems of networked actors. We parametrize an actors' rationality via their place in a social network and quantify system rationality via the average Jensen-Shannon divergence between the games Nash and logit quantal response equilibria. Previous work has argued that scale-free topologies maximize a system's overall rationality in this setup. Here we show that while, for certain games, it is true that increasing degree heterogeneity of complex networks enhances rationality, rationality-optimal configurations are not scale-free. For the Prisoner's Dilemma and Stag Hunt games, we provide analytic arguments complemented by numerical optimization experiments to demonstrate that core-periphery networks composed of a few dominant hub nodes surrounded by a periphery of very low degree nodes give strikingly smaller overall deviations from rationality than scale-free networks. Similarly, for the Battle of the Sexes and the Matching Pennies games, we find that the optimal network structure is also a core-periphery graph but with a smaller difference in the average degrees of the core and the periphery. These results provide insight on the interplay between the topological structure of socioecological systems and their collective cognitive behavior, with potential applications to understanding wealth inequality and the structural features of the network of global corporate control.

A Game-Theoretic Approach to Catching a Cheating Spouse

In this paper, we study a game model of marital cheating. The husband is the cheater and the wife is faithful. The husband’s cheating is either open or surreptitious. The wife can either ignore the cheating or catch her husband in the act of cheating. We first express the game of interest in matrix form. Second, we determine the best response functions of the two players. Third, we show that there exists a unique mixed-strategy Nash equilibrium in the game. Finally, we demonstrate the nexus between our marital cheating game and the prominent Matching Pennies game.

Cheating on Your Spouse: A Game-Theoretic Analysis

In this note we analyze a game of marital infidelity. The husband can either be faithful to or cheat on his wife. The wife can either monitor or not monitor her husband. We first determine the best response correspondences of the two players. Second, we explain why there is no pure-strategy Nash equilibrium in the game under study. Third, we show that there exists a unique mixed-strategy Nash equilibrium in the game. Finally, we demonstrate the nexus between our marital infidelity game and the prominent Matching Pennies game.

Identification of Biased Beliefs in Games of Incomplete Information Using Experimental Data

This paper studies the identification of players'; preferences and beliefs in empirical applications of discrete choice games using experimental data. The experiment comprises a set of games with similar features (e.g., two-player coordination games) where each game has different values for the players'; monetary payoffs. Each game can be interpreted as an experimental treatment group. The researcher assigns randomly subjects to play these games and observes the outcome of the game as described by the vector of players' actions. Data from this experiment can be described in terms of the empirical distribution of players' actions conditional on the treatment group. The researcher is interested in the nonparametric identification of players' preferences (utility function of money) and players' beliefs about the expected behavior of other players, without imposing restrictions such as unbiased or rational beliefs or a particular functional form for the utility of money. We show that the hypothesis of unbiased/rational beliefs is testable and propose a test of this null hypothesis. We apply our method to two sets of experiments conducted by Goeree and Holt (2001) and Heinemann, Nagel and Ockenfels (2009). Our empirical results suggest that in the matching pennies game, a player is able to correctly predict other player's behavior. In the public good coordination game, our test can reject the null hypothesis of unbiased beliefs when the payoff of the non-cooperative action is relatively low.

Testing the Quantal Response Hypothesis

This paper develops a formal test for consistency of players' behavior in a series of games with the quantal response equilibrium (QRE). The test exploits a characterization of the equilibrium choice probabilities in a QRE as the gradient of a convex function, which thus satisfies the cyclic monotonicity inequalities. Our testing procedure utilizes recent econometric results for moment inequality models. We assess our test using lab experimental data from a series of generalized matching pennies games. We reject the QRE hypothesis in the pooled data, but it cannot be rejected in the individual data for over half of the subjects.