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Battigalli (1997) has shown that in dynamic games with perfect information and without relevant ties, the forward induction concept of extensive-form rationalizability yields the backward induction outcome. In this paper we provide a new proof for this remarkable result, based on four steps. We first show that extensive-form rationalizability can be characterized by the iterated application of a special reduction operator, the strong belief reduction operator. We next prove that this operator satisfies a mild version of monotonicity, which we call monotonicity on reachable histories. This property is used to show that for this operator, every possible order of elimination leads to the same set of outcomes. We finally show that backward induction yields a possible order of elimination for the strong belief reduction operator. These four properties together imply Battigalli's theorem.

Phenomena like meat sharing in hunter-gatherers, self-sacrifice in intergroup conflicts, and voluntary contribution to public goods provision in laboratory experiments have led to the development of numerous theories on the evolution of altruistic in-group beneficial behavior in humans. Many of these theories abstract away from the effects of kinship on the incentives for public goods provision, though. Here, it is investigated analytically how genetic relatedness changes the incentive structure of that paradigmatic game which is conventionally used to model and experimentally investigate collective action problems: the linear public goods game. Using recent anthropological data sets on relatedness in 61 contemporary hunter-gatherer and horticulturalist societies the relevant parameters of this model are then estimated. It turns out that the kinship patterns observed in these societies substantially reduce the negative effect of increasing group size on incentives for public goods provision. It is suggested, therefore, that renewed attention should be given to inclusive fitness theory in the context of public goods provision also in sizable groups, because its explanatory power with respect to this central problem in the evolution of human cooperativeness and altruism might have been substantially underrated. JEL: B15, C72, D64, H41

All equilibrium concepts implicitly make a correct beliefs assumption, stating that a player believes that his opponents are correct about his first-order beliefs. In this paper we show that in many dynamic games of interest, this correct beliefs assumption may be incompatible with a very basic form of forward induction reasoning: the first two layers of extensive-form rationalizability ( Pearce, 1984 , Battigalli, 1997 , epistemically characterized by Battigalli and Siniscalchi, 2002 ). Hence, forward induction reasoning naturally leads us away from equilibrium reasoning. In the second part we classify the games for which equilibrium reasoning is consistent with this type of forward induction reasoning, and find that this class is very small.

We provide an existence result for the solution concept of generalized Nash equilibrium, which can be viewed as the direct incomplete information analogue of Nash equilibrium. Intuitively, a tuple consisting of a probability measure for every player on his choices and utility functions is a generalized Nash equilibrium, whenever some mutual optimality property is satisfied. This incomplete information solution concept is then epistemically characterized in a way that common belief in rationality is neither used nor implied. For the special case of complete information, an epistemic characterization of Nash equilibrium ensues as a corollary.

We provide comparable algorithms for the Dekel-Fudenberg procedure, iterated admissibility, proper rationalizability and full permissibility by means of the notions of likelihood orderings and preference restrictions. The algorithms model reasoning processes whereby each player's preferences over his own strategies are completed by eliminating likelihood orderings. We apply the algorithms for comparing iterated admissibility, proper rationalizability and full permissibility, and provide a sufficient condition under which iterated admissibility does not rule out properly rationalizable strategies. We also use the algorithms to examine an economically relevant strategic situation, namely a bilateral commitment bargaining game. Finally, we discuss the relevance of our algorithms for epistemic analysis.

Using a network approach we provide a characterization of a separating equilibrium for standard signaling games where the sender's payoff function is quasi-linear. Given a strategy of the sender, we construct a network where the node set and the length between two nodes are the set of the sender's type and the difference of signaling costs, respectively. Construction of a separating equilibrium is then equivalent to constructing the length between two nodes in the network under the condition that the response of the receiver is a node potential. When the set of the sender's type is a real interval, shortest path lengths are antisymmetric and a node potential is unique up to a constant. A strategy of the sender in a separating equilibrium is characterized by some differential equation with a unique solution. Our results can be readily applied to a broad range of economic situations, such as for example the standard job market signaling model of Spence, a model not captured by earlier papers.

Dynamic network flows, or network flows over time, constitute an important model for real-world situations in which steady states are unusual, such as urban traffic and the internet. These applications immediately raise the issue of analyzing dynamic network flows from a game-theoretic perspective. In this paper, we study dynamic equilibria in the deterministic fluid queuing model in single-source, single-sink networks—arguably the most basic model for flows over time. In the last decade, we have witnessed significant developments in the theoretical understanding of the model. However, several fundamental questions remain open. One of the most prominent ones concerns the price of anarchy, measured as the worst-case ratio between the minimum time required to route a given amount of flow from the source to the sink and the time a dynamic equilibrium takes to perform the same task. Our main result states that, if we could reduce the inflow of the network in a dynamic equilibrium, then the price of anarchy is bounded by a factor, parameterized by the longest path length that converges to [Formula: see text], and this is tight. This significantly extends a result by Bhaskar et al. (SODA 2011). Furthermore, our methods allow us to determine that the price of anarchy in parallel-link and parallel-path networks is exactly 4/3. Finally, we argue that, if a certain, very natural, monotonicity conjecture holds, the price of anarchy in the general case is exactly [Formula: see text].

In this paper, we consider a sequence of multi-prize all-pay auctions, where the contestants who exert the lowest e fforts in a round are eliminated. We analyze if and how the behavior of contestants is influenced by the possibility that strong rivals are eliminated along the way. The conditions under which behavior is not influenced are rather restrictive. For cases where these conditions are not met, we derive equilibria in a two-period model. We compare our equilibrium predictions to those of a static model and a two-stage contest where the sequential structure is not announced in the first round.

In this paper, we consider a sequence of multi-prize all-pay auctions, where the contestants who exert the lowest e fforts in a round are eliminated.We analyze if and how the behavior of contestants is influenced by the possibility that strong rivals are eliminated along the way. The conditions under which behavior is not influenced are rather restrictive. For cases where these conditions are not met, we derive equilibria in a two-period model. We compare our equilibrium predictions to those of a static model and a two-stage contest where the sequential structure is not announced in the first round.

In a local interaction model agents situated on a circle play bilateral prisoners’ dilemmas with their immediate neighbors and have three possible strategies: cooperate in all interactions (altruistic), defect in all interactions (egoistic), or cooperate with one immediate neighbor with probability 1=2 (partial cooperation). After each period the agents adopt the strategy with the highest average payoff in their observed local neighborhood (naïve imitation). The absorbing states of the process are outlined and analysed. There does not exist an absorbing state in which the partially cooperative strategy coexists with any of the other strategies. The partially cooperative strategy limits the diffusion of altruistic behavior in the population. Even though clustering of altruists is generally beneficial for sustaining altruism, relatively big groups of altruists at the onset actually enable the spread of the partially cooperative strategy.

In a local interaction model agents situated on a circle play bilateral prisoners’ dilemmas with their immediate neighbors and have three possible strategies: cooperate in all interactions (altruistic), defect in all interactions (egoistic), or cooperate with one immediate neighbor with probability 1=2 (partial cooperation). After each period the agents adopt the strategy with the highest average payoff in their observed local neighborhood (naïve imitation). The absorbing states of the process are outlined and analysed. There does not exist an absorbing state in which the partially cooperative strategy coexists with any of the other strategies. The partially cooperative strategy limits the diffusion of altruistic behavior in the population. Even though clustering of altruists is generally beneficial for sustaining altruism, relatively big groups of altruists at the onset actually enable the spread of the partially cooperative strategy.

In a local interaction model agents situated on a circle play bilateral prisoners’ dilemmas with their immediate neighbors and have three possible strategies: cooperate in all interactions (altruistic), defect in all interactions (egoistic), or cooperate with one immediate neighbor with probability 1=2 (partial cooperation). After each period the agents adopt the strategy with the highest average payoff in their observed local neighborhood (naïve imitation). The absorbing states of the process are outlined and analysed. There does not exist an absorbing state in which the partially cooperative strategy coexists with any of the other strategies. The partially cooperative strategy limits the diffusion of altruistic behavior in the population. Even though clustering of altruists is generally beneficial for sustaining altruism, relatively big groups of altruists at the onset actually enable the spread of the partially cooperative strategy.

In this paper we focus on stochastic games with finitely many states and actions. For this setting we study the epistemic concept of common belief in future rationality, which is based on the condition that players always believe that their opponents will choose rationally in the future. We distinguish two different versions of the concept—one for the discounted case with a fixed discount factor $$\delta ,$$ and one for the case of uniform optimality, where optimality is required for all discount factors close enough to 1” . We show that both versions of common belief in future rationality are always possible in every stochastic game, and always allow for stationary optimal strategies. That is, for both versions we can always find belief hierarchies that express common belief in future rationality, and that have stationary optimal strategies. We also provide an epistemic characterization of subgame perfect equilibrium for two-player stochastic games, showing that it is equivalent to mutual belief in future rationality together with some “correct beliefs assumption”.

We study games with almost perfect information and an infinite time horizon. In such games, at each stage, the players simultaneously choose actions from finite action sets, knowing the actions chosen at all previous stages. The payoff of each player is a function of all actions chosen during the game. We define and examine the new condition of individual upper semicontinuity on the payoff functions, which is weaker than upper semicontinuity. We prove that a game with individual upper semicontinuous payoff functions admits a subgame perfect ϵ-equilibrium for every ϵ > 0, in eventually pure strategy profiles.

We study games with almost perfect information and an infinite time horizon. In such games, at each stage, the players simultaneously choose actions from finite action sets, knowing the actions chosen at all previous stages. The payoff of each player is a function of all actions chosen during the game. We define and examine the new condition of individual upper semicontinuity on the payoff functions, which is weaker than upper semicontinuity. We prove that a game with individual upper semicontinuous payoff functions admits a subgame perfect ϵ-equilibrium for every ϵ > 0, in eventually pure strategy profiles.

We study games with almost perfect information and an infinite time horizon. In such games, at each stage, the players simultaneously choose actions from finite action sets, knowing the actions chosen at all previous stages. The payoff of each player is a function of all actions chosen during the game. We define and examine the new condition of individual upper semicontinuity on the payoff functions, which is weaker than upper semicontinuity. We prove that a game with individual upper semicontinuous payoff functions admits a subgame perfect ϵ-equilibrium for every ϵ > 0, in eventually pure strategy profiles.

Existing models of regret aversion assume that individuals can make an ex-post comparison between their choice and a foregone alternative. Yet in many situations such a comparison can be made only if someone else chose the alternative option. We develop a model where regret-averse agents must decide between the status quo and a new risky option that outperforms the status quo in expectation, and learn the outcome of the risky option, if unchosen, with a probability that depends on the choices of others. This turns what was previously a series of single-person decision problems into a coordination game. Most notably, regret can facilitate coordination on the status quo { an action that would not be observed if the agents were acting in isolation or had standard preferences. We experimentally test the model and find that regret-averse agents behave as predicted by our theory.

We present a general existence result for a type of equilibrium in normal-form games. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a justifiable equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a justifiable equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from Vasquez (2017), but is more direct as it does not make use of countably additive representations of finitely additive measures by Yosida and Hewitt (1952).

We present a general existence result for a type of equilibrium in normal-form games. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a justifiable equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a justifiable equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from Vasquez (2017), but is more direct as it does not make use of countably additive representations of finitely additive measures by Yosida and Hewitt (1952).

We present a general existence result for a type of equilibrium in normal-form games. We consider nonzero-sum normal-form games with an arbitrary number of players and arbitrary action spaces. We impose merely one condition: the payoff function of each player is bounded. We allow players to use finitely additive probability measures as mixed strategies. Since we do not assume any measurability conditions, for a given strategy profile the expected payoff is generally not uniquely defined, and integration theory only provides an upper bound, the upper integral, and a lower bound, the lower integral. A strategy profile is called a justifiable equilibrium if each player evaluates this profile by the upper integral, and each player evaluates all his possible deviations by the lower integral. We show that a justifiable equilibrium always exists. Our equilibrium concept and existence result are motivated by Vasquez (2017), who defines a conceptually related equilibrium notion, and shows its existence under the conditions of finitely many players, separable metric action spaces and bounded Borel measurable payoff functions. Our proof borrows several ideas from Vasquez (2017), but is more direct as it does not make use of countably additive representations of finitely additive measures by Yosida and Hewitt (1952).

Farsighted deviations are based on agents' abilities to compare the outcome of a farsighted deviation to the status quo. However, agents do not account for deviations by others in case they do not change the status quo; so, they are not fully farsighted. We use extended expectation functions to capture a coalition's belief about subsequent moves of other coalitions in both cases. We provide three stability and optimality axioms on coalition behavior and show that an expectation function satisfies these axioms if and only if it corresponds to an equilibrium of the abstract game that is stable with respect to coalitional deviations. We provide applications of our solution for games in characteristic function form and matching problems.

Farsighted deviations are based on agents' abilities to compare the outcome of a farsighted deviation to the status quo. However, agents do not account for deviations by others in case they do not change the status quo; so, they are not fully farsighted. We use extended expectation functions to capture a coalition's belief about subsequent moves of other coalitions in both cases. We provide three stability and optimality axioms on coalition behavior and show that an expectation function satisfies these axioms if and only if it corresponds to an equilibrium of the abstract game that is stable with respect to coalitional deviations. We provide applications of our solution for games in characteristic function form and matching problems.

We consider two versions of a Bertrand duopoly with asymmetric costs and homogeneous goods. They differ in whether predatory pricing is allowed. For each version, we derive the Myopic Stable Set in pure strategies as introduced by Demuynck, Herings, Saulle, and Seel (2017). We contrast our prediction to the prediction of Nash Equilibrium in mixed strategies.

We consider two versions of a Bertrand duopoly with asymmetric costs and homogeneous goods. They differ in whether predatory pricing is allowed. For each version, we derive the Myopic Stable Set in pure strategies as introduced by Demuynck, Herings, Saulle, and Seel (2017). We contrast our prediction to the prediction of Nash Equilibrium in mixed strategies.

We consider two versions of a Bertrand duopoly with asymmetric costs and homogeneous goods. They differ in whether predatory pricing is allowed. For each version, we derive the Myopic Stable Set in pure strategies as introduced by Demuynck, Herings, Saulle, and Seel (2017). We contrast our prediction to the prediction of Nash Equilibrium in mixed strategies.

We consider a model of stochastic evolution under general noisy best-response protocols, allowing the probabilities of suboptimal choices to depend on their payoff consequences. Our analysis focuses on behavior in the small noise double limit: we first take the noise level in agents' decisions to zero, and then take the population size to infinity. We show that in this double limit, escape from and transitions between equilibria can be described in terms of solutions to continuous optimal control problems. These are used in turn to characterize the asymptotics of the stationary distribution, and so to determine the stochastically stable states. We use these results to perform a complete analysis of evolution in three-strategy coordination games that satisfy the marginal bandwagon property and that have an interior equilibrium, with agents following the logit choice rule.

Caution is an integral part of many solution concepts in traditional game theory and is commonly modelled using lexicographic beliefs. We show here that lexicographic beliefs lack the expressive power to model caution once we extend traditional games to psychological games. Quantification of the relation of ‘deeming an event infinitely more likely than another event’ is necessary, which can be accomplished by using non-standard beliefs.

Caution is an integral part of many solution concepts in traditional game theory and is commonly modelled using lexicographic beliefs. We show here that lexicographic beliefs lack the expressive power to model caution once we extend traditional games to psychological games. Quantification of the relation of ‘deeming an event infinitely more likely than another event’ is necessary, which can be accomplished by using non-standard beliefs.

Caution is an integral part of many solution concepts in traditional game theory and is commonly modelled using lexicographic beliefs. We show here that lexicographic beliefs lack the expressive power to model caution once we extend traditional games to psychological games. Quantification of the relation of ‘deeming an event infinitely more likely than another event’ is necessary, which can be accomplished by using non-standard beliefs.

This paper studies adaptive learning in the class of weighted network games. This class of games includes applications like research and development within interlinked firms, crime within social networks, the economics of pollution, and defense expenditures within allied nations. We show that for every weighted network game, the set of pure Nash equilibria is non-empty and, generically, finite. Pairs of players are shown to have jointly profitable deviations from interior Nash equilibria. If all interaction weights are either non-negative or non-positive, then Nash equilibria are Pareto inefficient. We show that quite general learning processes converge to a Nash equilibrium of a weighted network game if every player updates with some regularity.

This paper studies adaptive learning in the class of weighted network games. This class of games includes applications like research and development within interlinked firms, crime within social networks, the economics of pollution, and defense expenditures within allied nations. We show that for every weighted network game, the set of pure Nash equilibria is non-empty and, generically, finite. Pairs of players are shown to have jointly profitable deviations from interior Nash equilibria. If all interaction weights are either non-negative or non-positive, then Nash equilibria are Pareto inefficient. We show that quite general learning processes converge to a Nash equilibrium of a weighted network game if every player updates with some regularity.

This paper studies adaptive learning in the class of weighted network games. This class of games includes applications like research and development within interlinked firms, crime within social networks, the economics of pollution, and defense expenditures within allied nations. We show that for every weighted network game, the set of pure Nash equilibria is non-empty and, generically, finite. Pairs of players are shown to have jointly profitable deviations from interior Nash equilibria. If all interaction weights are either non-negative or non-positive, then Nash equilibria are Pareto inefficient. We show that quite general learning processes converge to a Nash equilibrium of a weighted network game if every player updates with some regularity.

We conducted a controlled laboratory experiment to understand how and what kind of information triggers adoption and abandonment of different decision-making principles in games. We consider three types of decision-making principles: best-response, payoff-based learning, and imitation. Our focus is on Cournot contests, where Nash equilibrium is located between social optimum and social pessimum. Subjects start in a low-information environment, where only payoff-based learning is feasible, and end up with full information, where all types of decision-making rules apply. Treatments vary with respect to the order with which information is revealed as the game is repeated. Our study is designed to address three main questions. (1) The 'marginal' effect of information: Which new bits of information trigger which principle? (2) The 'additive' effect: How does the history of previously available information affect (1)? (3) The 'substitution' effect: How are decision-making principles abandoned and adopted? Thus, we establish a novel link from micro-heuristics to various resulting macro-dynamics of play, which converge either toward Nash equilibrium or toward a socially inferior zero-profit outcome.

We study the effect of noise due to exogenous information distortions in the context of Bayesian persuasion. We first characterize the optimal signal in the prosecutor-judge game from Kamenica and Gentzkow (2011) with a noisy and strongly symmetric communication channel and show that the sender's payoff increases in the number of messages. This implies that, with exogenous noise, the sender prefers to complicate communication. Then, we establish necessary and sufficient conditions for the sender's payoff to weakly increase in the Blackwell-informativeness of the noise channel when the message space and the channel are binary. The reason why a sender may benefit from additional noise is that a garbling may alter the noise structure. Subsequently, we provide sufficient conditions that extend this result to channels of arbitrary cardinality. Finally, we introduce a procedure of making a communication channel more complex and prove that increased complexity benefits the sender.

We conducted a controlled laboratory experiment to understand how and what kind of information triggers adoption and abandonment of different decision-making principles in games. We consider three types of decision-making principles: best-response, payoff-based learning, and imitation. Our focus is on Cournot contests, where Nash equilibrium is located between social optimum and social pessimum. Subjects start in a low-information environment, where only payoff-based learning is feasible, and end up with full information, where all types of decision-making rules apply. Treatments vary with respect to the order with which information is revealed as the game is repeated. Our study is designed to address three main questions. (1) The 'marginal' effect of information: Which new bits of information trigger which principle? (2) The 'additive' effect: How does the history of previously available information affect (1)? (3) The 'substitution' effect: How are decision-making principles abandoned and adopted? Thus, we establish a novel link from micro-heuristics to various resulting macro-dynamics of play, which converge either toward Nash equilibrium or toward a socially inferior zero-profit outcome.

The paper provides mathematical foundations for a homeomorphism theorem à la Mertens and Zamir (1985) when the space of belief hierarchies of an agent has the uniform topology rather than the product topology. The Borel σ-algebra for the uniform topology being unsuitable, the theorem relies on the product σ-algebra but defines the topology of weak convergence on the space of measures on this σ-algebra with reference to the uniform topology on the underlying space. For a countable product of complete separable metric spaces, the paper shows that this topology on the space of measures on the product σ-algebra is metrizable by the Prohorov metric. The projection mapping from such measures to sequences of measures on the first ℓ factors, ℓ=1,2,..., is a homeomorphism if the range of this mapping is also given a uniform metric.

We conducted a controlled laboratory experiment to understand how and what kind of information triggers adoption and abandonment of different decision-making principles in games. We consider three types of decision-making principles: best-response, payoff-based learning, and imitation. Our focus is on Cournot contests, where Nash equilibrium is located between social optimum and social pessimum. Subjects start in a low-information environment, where only payoff-based learning is feasible, and end up with full information, where all types of decision-making rules apply. Treatments vary with respect to the order with which information is revealed as the game is repeated. Our study is designed to address three main questions. (1) The 'marginal' effect of information: Which new bits of information trigger which principle? (2) The 'additive' effect: How does the history of previously available information affect (1)? (3) The 'substitution' effect: How are decision-making principles abandoned and adopted? Thus, we establish a novel link from micro-heuristics to various resulting macro-dynamics of play, which converge either toward Nash equilibrium or toward a socially inferior zero-profit outcome.

We study unanimity bargaining on the division of a surplus in the presence of monotonicity constraints. The monotonicity constraints specify a complete order on the players, which has to be respected by the shares in the surplus the players obtain in any bargaining outcome. A player higher in the order should not receive a lower share of the surplus. We analyze the resulting subgame perfect equilibria in stationary strategies and show that they are characterized by the simpler notion of bargaining equilibrium. Bargaining equilibria are shown to be unique and to have the property that players ranked strictly higher obtain strictly higher shares in the surplus. The key question is whether the bargaining advantage of a higher-ranked player persists when the probability of breakdown of bargaining tends to zero. We argue that such is not the case by showing that bargaining equilibria have a unique limit equal to an equal division of the surplus. It then follows that the limit also coincides with the Nash bargaining solution for this problem.

We study perfect information games played by an infinite sequence of players, each acting only once in the course of the game. We introduce a class of frequency-based minority games and show that these games admit no subgame perfect ϵ-equilibrium for small positive values of ϵ. Furthermore we derive a number of sufficient conditions to guarantee existence of subgame perfect ϵ-equilibrium.