Using turn taking to achieve intertemporal cooperation and symmetry in infinitely repeated 2 × 2 games

Turn taking is observed in many field and laboratory settings captured by various widely studied 2 × 2 games. This article develops a repeated game model that allows us to systematically investigate turn-taking behavior in many 2 × 2 games, including the battle of the sexes, the game of chicken, the game of common-pool-resources assignment, and a particular version of the prisoners’ dilemma. We consider the “turn taking with independent randomizations” (TTIR) strategy that achieves three objectives: (a) helping the players reach the turn-taking path, (b) resolving the question of who takes the good turn first, and (c) deterring defection. We determine conditions under which there exists a unique TTIR strategy profile that can be supported as a subgame-perfect equilibrium. We also show that there exist conditions under which an increase in the “degree of conflict” of the stage game leads to a decrease in the expected number of periods in reaching the turn-taking path.     

Ellsberg games

In the standard formulation of game theory, agents use mixed strategies in the form of objective and probabilistically precise devices to conceal their actions. We introduce the larger set of probabilistically imprecise devices and study the consequences for the basic results on normal form games. While Nash equilibria remain equilibria in the extended game, there arise new Ellsberg equilibria with distinct outcomes, as we illustrate by negotiation games with three players. We characterize Ellsberg equilibria in two-person conflict and coordination games. These equilibria turn out to be related to experimental deviations from Nash equilibrium play.     

No Switchbacks: Rethinking Aspiration-Based Dynamics in the Ultimatum Game

Aspiration-based evolutionary dynamics have recently been used to model the evolution of fair play in the ultimatum game showing that incredible threats to reject low offers persist in equilibrium. We focus on two extensions of this analysis: we experimentally test whether assumptions about agent motivations (aspiration levels) and the structure of the game (binary strategy space) reflect actual play, and we examine the problematic assumption embedded in the standard replicator dynamic that unhappy agents who switch strategies may return to a rejected strategy without exploring other options. We find that the resulting “no switchback” dynamic predicts the evolution of play better than the standard dynamic and that aspirations are a significant motivator for our participants. In the process, we also construct and analyze a variant of the ultimatum game in which players can adopt conditional (on their induced aspirations) strategies.     

Interpersonal comparisons of utility in bargaining: evidence from a transcontinental ultimatum game

This article presents the experimental results of a “Transcontinental Ultimatum Game” implemented between India and France. We use a standard ultimatum game, but in one treatment, Indian subjects made offers to French subjects (ItoF treatment) and, in another treatment, French subjects made offers to Indian subjects (FtoI treatment). We observed that FtoI treatment bargaining mostly ended up with unequal splits of money in favor of French, while nearly equal splits were the most frequent outcome in ItoF treatment interactions. The experimental results are organized through a standard social reference model, modified for taking into account the different marginal value of money for bargainers. In our model, bargaining is driven by relative standing comparisons between players, occurring in terms of real earnings (that is monetary earnings corrected for a purchasing power factor) obtained in the game. The norm of equity behind the equalization of real earnings is called local equity norm, and contrasted to a global equity norm which would encompass the wealth of players beyond the game. According to what we observed, no beyond-game concern seems to be relevantly endorsed by subjects.     

Endogenizing the order of moves in matrix games

Players often have flexibility in when they move and thus whether a game is played simultaneously or sequentially may be endogenously determined. For 2 × 2 games, we analyze this using an extended game. In a stage prior to actual play, players choose in which of two periods to move. A player moving at the first opportunity knows when his opponent will move. A player moving at the second turn learns the first mover's action. If both select the same turn, they play a simultaneous move subgame.If both players have dominant strategies in the basic game, equilibrium payoffs in the basic and extended games are identical. If only one player has a dominant strategy or if the unique equilibrium in the basic game is in mixed strategies, then the extended game equilibrium payoffs differ if and only if some pair of pure strategies Pareto dominates the basic game simultaneous play payoffs. If so, sequential play attains the Pareto dominating payoffs. The mixed strategy equilibrium occurs only when it is not Pareto dominated by some pair of pure strategies.In an alternative extended game, players cannot observe delay by opponents at the first turn. Results for 2×2 games are essentially the same as with observable delay, differing only when only one player has a dominant strategy.     

Characterizations of the β- and the Degree Network Power Measure

A symmetric network consists of a set of positions and a set of bilateral links between these positions. For every symmetric network we define a cooperative transferable utility game that measures the “power” of each coalition of positions in the network. Applying the Shapley value to this game yields a network power measure, the β-measure, which reflects the power of the individual positions in the network. Applying this power distribution method iteratively yields a limit distribution, which turns out to be proportional to the well-known degree measure. We compare the β-measure and degree measure by providing characterizations, which differ only in the normalization that is used.      

Economic Darwinism

We define an evolutionary process of “economic Darwinism” for playing the field, symmetric games. The process captures two forces. One is “economic selection”: if current behavior leads to payoff differences, behavior yielding lowest payoff has strictly positive probability of being replaced by an arbitrary behavior. The other is “mutation”: any behavior has at any point in time a strictly positive, very small probability of shifting to an arbitrary behavior. We show that behavior observed frequently is in accordance with “evolutionary equilibrium”, a static equilibrium concept suggested in the literature. Using this result, we demonstrate that generally under positive (negative) externalities, economic Darwinism implies even more under- (over-)activity than does Nash equilibrium.     

Overconfidence in tournaments: evidence from the field

This paper uses a field survey to investigate the quality of individuals’ beliefs of relative performance in tournaments. We consider two field settings, poker and chess, which differ in the degree to which luck is a factor and also in the information that players have about the ability of the competition. We find that poker players’ forecasts of relative performance are random guesses with an overestimation bias. Chess players also overestimate their relative performance but make informed guesses. We find support for the “unskilled and unaware hypothesis” in chess: high-skilled chess players make better forecasts than low-skilled chess players. Finally, we find that chess players’ forecasts of relative performance are not efficient.     

Information transparency and equilibrium selection in coordination games: an experimental study

We experimentally investigate the role of information transparency for equilibrium selection in stag hunt coordination games. These games can be transformed from a prisoner’s dilemma game by introducing a centralized reward or punishment scheme. We aim to explore the impact of the disclosure of information on how final payoffs are derived on players’ incentive to coordinate on the payoff-dominant equilibrium. We find that such information disclosure significantly increases the tendency of players to play the payoff-dominant strategy and reduces the occurrence of coordination failure. The mechanism works directly through the positive impact of disclosure on the saliency of the payoff-dominant equilibrium, and indirectly through the positive influence of disclosure on players’ belief about the likelihood of cooperation by the opponent.     

Institutions matter! Why the Herder Problem is not a Prisoner’s Dilemma

In the game theory literature, Garrett Hardin’s famous allegory of the “tragedy of the commons” has been modeled as a variant of the Prisoner’s Dilemma, labeled the Herder Problem (or, sometimes, the Commons Dilemma). This brief paper argues that important differences in the institutional structures of the standard Prisoner’s Dilemma and Herder Problem render the two games different in kind. Specifically, institutional impediments to communication and cooperation that ensure a dominant strategy of defection in the classic Prisoner’s Dilemma are absent in the Herder Problem. Their absence does not ensure that players will achieve a welfare-enhancing, cooperative solution to the Herders Problem, but does create far more opportunity for players to alter the expected payoffs through cooperative arrangements. In a properly modeled Herder Problem—along the lines of an assurance game—defection would not always be the dominant strategy. Consequently, the Herder Problem is not in the nature of a Prisoner’s Dilemma.     

A Philosophical Foundation of Non-Additive Measure and Probability

In this paper, non-additivity of a set function is interpreted as a method to express relations between sets which are not modeled in a set theoretic way. Drawing upon a concept called “quasi-analysis” of the philosopher Rudolf Carnap, we introduce a transform for sets, functions, and set functions to formalize this idea. Any image-set under this transform can be interpreted as a class of (quasi-)components or (quasi-)properties representing the original set. We show that non-additive set functions can be represented as signed σ-additive measures defined on sets of quasi-components. We then use this interpretation to justify the use of non-additive set functions in various applications like for instance multi criteria decision making and cooperative game theory. Additionally, we show exemplarily by means of independence, conditioning, and products how concepts from classical measure and probability theory can be transfered to the non-additive theory via the transform.     

Delay in a bargaining game with contracts

In a multilateral bargaining game where a proposer and responders can set up a “principal–agent” relationship by means of binding cash-offer contracts, we show that there is a Markov SPE with a delay in reaching an agreement. We also show that all the individually rational and efficient payoffs can be supported by SPE.      

Perceptron versus automaton in the finitely repeated prisoner’s dilemma

We study the finitely repeated prisoner’s dilemma in which the players are restricted to choosing strategies which are implementable by a machine with a bound on its complexity. One player has to use a finite automaton while the other player has to use a finite perceptron. Some examples illustrate that the sets of strategies which are induced by these two types of machines are different and not ordered by set inclusion. Repeated game payoffs are evaluated according to the limit of means. The main result establishes that a cooperation at almost all stages of the game is an equilibrium outcome if the complexity of the machines the players may use is limited enough and if the length T of the repeated game is sufficiently large. This result persists when more than T states are allowed in the player’s automaton. We further consider a variant of the model in which the two players are restricted to choosing strategies which are implementable by perceptrons and prove that the players can cooperate at most of the stages provided that the complexity of their perceptrons is sufficiently reduced.     

Strategic behavior under partial cooperation

We investigate how a group of players might cooperate with each other within the setting of a non-cooperative game. We pursue two notions of partial cooperative equilibria that follow a modification of Nash’s best response rationality rather than a core-like approach. Partial cooperative Nash equilibrium treats non-cooperative players and the coalition of cooperators symmetrically, while the notion of partial cooperative leadership equilibrium assumes that the group of cooperators has a first-mover advantage. We prove existence theorems for both types of equilibria. We look at three well-known applications under partial cooperation. In a game of voluntary provision of a public good we show that our two new equilibrium notions of partial cooperation coincide. In a modified Cournot oligopoly, we identify multiple equilibria of each type and show that a non-cooperator may have a higher payoff than a cooperator. In contrast, under partial cooperation in a symmetric Salop City game, a cooperator enjoys a higher return.     

Guaranteed renewability uniquely prevents adverse selection in individual health insurance

New models of multi-period insurance show that health insurance buyers can be protected against changes in premiums from health shocks associated with chronic conditions by the addition of “guaranteed renewability” provisions. These models assume that a buyer’s risk level in every time period is observed by all insurers. They also require a premium sequence that is “front-loaded,” which may be costly to buyers if capital markets are imperfect. We relax the common knowledge feature of the model by assuming that a person’s risk in any time period is known only by that individual and the current insurer. One might suspect that a premium sequence with higher later period premiums would be incentive compatible because low risks will have less desirable offerings from alternative insurers. However, we show that generally, only the original premium schedule is incentive compatible, and attempts to alter front-loading will not be an equilibrium.     

Information-driven coordination: experimental results with heterogeneous individuals

We study experimentally a coordination game with N heterogeneous individuals under different information treatments. We explore the effects of information on the emergence of Pareto-efficient outcomes, by means of a gradual decrease of the information content provided to the players in successive experiments. We observe that successful coordination is possible with private information alone, although not on a Pareto-optimal equilibrium. Reinforcement-based learning models reproduce the qualitative trends of the experimental results.     

Cycles versus equilibrium in evolutionary games

Mixed-strategy equilibria are typically rather unstable in evolutionary game theory. “Monocyclic” games, such as Rock–Paper–Scissors, have only mixed equilibria, some of which are “stable” in the sense that sequential best replies lead to them; yet, even these games are prone to stable cycles under discrete-time simultaneous best replies, giving an unusual equilibrium-selection problem. This article analyzes such games in a random-utility setting where changing strategies is costly, and the speed of the dynamic is, thus, endogenous. The stochastically stable outcome is determined by the cost of switching strategies; when switching costs are high, mixed equilibria are selected, whereas when switching costs are low, cycles are selected.     

History as a coordination device

Coordination games often have multiple equilibria. The selection of equilibrium raises the question of belief formation: how do players generate beliefs about the behavior of other players? This article takes the view that the answer lies in history, that is, in the outcomes of similar coordination games played in the past, possibly by other players. We analyze a simple model in which a large population plays a game that exhibits strategic complementarities. We assume a dynamic process that faces different populations with such games for randomly selected values of a parameter. We introduce a belief formation process that takes into account the history of similar games played in the past, not necessarily by the same population. We show that when history serves as a coordination device, the limit behavior depends on the way history unfolds, and cannot be determined from a-priori considerations.     

Infinity in the lab. How do people play repeated games?

We introduce a novel mechanism to eliminate endgame effects in repeated prisoner’s dilemma experiments. In the main phase of a supergame our mechanism generates more persistent cooperation than finite horizon or random continuation rules. Moreover, we find evidence for cooperation-enhancing “active/reactive” strategies which concentrate in the initial phase of a supergame as subjects gain experience.     

Self-protection and insurance with interdependencies

We study optimal investment in self-protection of insured individuals when they face interdependencies in the form of potential contamination from others. If individuals cannot coordinate their actions, then the positive externality of investing in self-protection implies that, in equilibrium, individuals underinvest in self-protection. Limiting insurance coverage through deductibles or selling “at-fault” insurance can partially internalize this externality and thereby improve individual and social welfare.