Games

This article is an overview of recent progress on a theory of games, whose payoffs are probability distributions rather than real numbers, and which have their equilibria defined and computed over a (suitably restricted yet dense) set of distributions. While the classical method of defining game models with real-valued utility functions has proven strikingly successful in many domains, some use cases from the security area revealed shortcomings of the classical real-valued game models. These issues motivated the use of probability distributions as a more complex object to express revenues. The resulting class of games displays a variety of phenomena not encountered in classical games, such as games that have continuous payoff functions but still no equilibrium, or games that are zero-sum but for which fictitious play does not converge. We discuss suitable restrictions of how such games should be defined to allow the definition of equilibria, and show the notion of a lexicographic Nash equilibrium, as a proposed solution concept in this generalized class of games. Full article

In this paper, we present the results of an experiment conducted in Italy on trusting behaviour. Our subjects participated in a trust game and filled in a questionnaire on trust and trustworthiness based on the attitudinal questions reported in the European Value Survey. The aims of the research are twofold. Firstly, using the experimental dataset, we construct two measures of trustworthiness among all recipients in the experiment, one based on the questionnaires’ answers and another based on the strategy method. We then compare the ex-ante behavioural decision to trust (before participants are allocated to a group) with the ex-post decision to trust (after participants are allocated to a group and trustors are informed of the level of trustworthiness of all the recipients who have been randomly allocated to each group). Our main finding is that trust strongly varies once the information is disclosed to trustors. The effect on trust is greater when the strategy method is used. Secondly, we compare the behavioural measure of trust with the attitudinal measures of trust and trustworthiness and find that there is only a weak correlation between the two measures, thus confirming, for the Italian case, similar findings in country-based studies on trust. Full article

Let ${\mathsf{\Phi }}_n$ be the set of the binary strategy-proof social choice functions referred to a group of n voters who are allowed to declare indifference between the alternatives. We provide a recursive way to obtain the set ${\mathsf{\Phi }}_{n+1}$ from the set ${\mathsf{\Phi }}_n$. Computing the cardinalities $|{\mathsf{\Phi }}_n|$ presents difficulties as the computation of the Dedekind numbers. The latter give the analogous number of social choice functions when only strict preferences are admitted. A comparison is given for the known values. Based on our results, we present a graphical description of the binary strategy-proof social choice functions in the case of three voters. Full article

This paper studies how the cost of delay and voting order affect agents’ decisions in a unanimity voting mechanism. Specifically, we consider two-voter conclaves with commonly known preferences over two alternatives, the cost of delay, and the following novelty: each voter has a subjective deadline—a moment in time when he/she prefers immediate agreement on any alternative, rather than future agreement on his/her most-preferred alternative. Our key finding shows that patience is not necessarily a main attribute of strategic advantage. When the first voter is the same at every stage, this voter will obtain his/her preferred alternative, even if he/she is the least patient one. However, this first movement advantage disappears when agents alternate as the first voter of each stage: in this case, the most patient voter always wins. Full article

The black box method was developed as an “asocial control” to allow for payoff-based learning while eliminating social responses in repeated public goods games. Players are told they must decide how many virtual coins they want to input into a virtual black box that will provide uncertain returns. However, in truth, they are playing with each other in a repeated social game. By “black boxing” the game’s social aspects and payoff structure, the method creates a population of self-interested but ignorant or confused individuals that must learn the game’s payoffs. This low-information environment, stripped of social concerns, provides an alternative, empirically derived null hypothesis for testing social behaviours, as opposed to the theoretical predictions of rational self-interested agents (Homo economicus). However, a potential problem is that participants can unwittingly affect the learning of other participants. Here, we test a solution to this problem in a range of public goods games by making participants interact, unknowingly, with simulated players (“computerised black box”). We find no significant differences in rates of learning between the original and the computerised black box, therefore either method can be used to investigate learning in games. These results, along with the fact that simulated agents can be programmed to behave in different ways, mean that the computerised black box has great potential for complementing studies of how individuals and groups learn under different environments in social dilemmas. Full article

We consider the problem of allocating heterogeneous objects to agents with money, where the number of agents exceeds that of objects. Each agent can receive at most one object, and some objects may remain unallocated. A bundle is a pair consisting of an object and a payment. An agent’s preference over bundles may not be quasi-linear, which exhibits income effects or reflects borrowing costs. We investigate the class of rules satisfying one of the central properties of fairness in the literature, egalitarian-equivalence, together with the other desirable properties. We propose (i) a novel class of rules that we call the independent second-prices rules with variable constraints and (ii) a novel condition on constraints that we call respecting the valuation coincidence. Then, we establish that the independent second-prices rule with variable constraints that respects the valuation coincidence is the only rule satisfying egalitarian-equivalence, strategy-proofness, individual rationality, and no subsidy for losers. Our characterization result implies that in the case of three or more agents, there are few opportunities for agents to receive objects under a rule satisfying egalitarian-equivalence and the other desirable properties, which highlights the strong tension between egalitarian-equivalence and efficiency. In contrast, in the case of two agents and a single object, egalitarian-equivalence is compatible with efficiency. Full article

We experimentally investigated the effects of the possibility of taking in the dictator game and the choices of passive players between the dictator game and the taking game on the distribution decisions of active players. Our main findings support our hypothesis: when the dictator game is not exogenously given but chosen by the receivers (or passive players), this makes them accountable, which leads to less giving by dictators. We also conducted an online survey to gain further insights about our experimental results. Survey participants predicted most of the observed behavior in the experiment and explained the factors that might have driven the predicted behavior using reasoning similar to ours. Our results provide a new perspective for the dependence of giving in the dictator game on contextual factors. Full article

Revision game is a very recent advance in dynamic game theory and it can be used to analyze the trading in the pre-opening stock market. In such games, players prepare actions that will be implemented at a given deadline, before which they may have opportunities to revise actions. For the first time, we study the role of the deadline in revision games, which is the core component that distinguishes revision games from classic games. We introduce the deadline distribution into revision game model and characterize the sufficient and necessary condition for players’ strategies to constitute an equilibrium. The equilibrium strategy with respect to the deadline uncertainty is given by a simple differential equation set. Governed by this differential equation set, players initially fully cooperate, and the cooperation level decreases as time progresses. The uncertainty has a great impact on players’ behavior. As the uncertainty increases, players become more risk averse, in the sense that they prefer lower mutual cooperation rate rather than higher payoff with higher uncertainty. Specifically, they will not stay in full cooperation for a long time, while after they deviate from the full cooperation, they adjust their plans more slowly and cautiously. The deadline uncertainty can improve the competition and avoid collusion in games, which could be utilized for auction design and pre-opening stock market regulations. Full article

We argue in favor of a departure from the equilibrium approach in game theory towards the less ambitious goal of describing only the actual behavior of rational players. The notions of Nash equilibrium and its refinements require a specification of the players’ choices and beliefs not only along the equilibrium play but also at counterfactual histories. We discuss an alternative—counterfactual-free—approach that focuses on choices and beliefs along the actual play, while being silent on choices and beliefs at unreached histories. Such an approach was introduced in an earlier paper that considered only perfect-information games. Here we extend the analysis to general extensive-form games (allowing for imperfect information) and put forward a behavioral notion of self-confirming play, which is close in spirit to the literature on self-confirming equilibrium. We also extend, to general extensive-form games, the characterization of rational play that is compatible with pure-strategy Nash equilibrium. Full article

We consider a market setting where a consumer holds either a naive or sophisticated perception of their preference over products. We introduce the concept of a cognitive equilibrium, in which the consumer can transition between the cognitive states of naiveté and sophistication depending on the degree of exploitation in the market. We compare market outcomes under monopoly and competition. While competition unambiguously improves market outcomes when the consumer’s cognitive state is exogenous, it can strictly lower gains from trade when cognitive states are endogenously determined. Full article

Almost every supplier faces uncertain and time-varying demand. E-commerce and online shopping have given suppliers unprecedented access to data on customers’ behavior, which sheds light on demand uncertainty. The main purpose of this research project is to provide an analytic tool for decentralized supply channel members to devise optimal long-term (multi-period) supply, pricing, and timing strategies while catering to stochastic demand in a diverse set of market scenarios. Despite its ubiquity in potential applications, the time-dependent channel optimization problem in its general form has received limited attention in the literature due to its complexity and the highly nested structure of its ensuing equilibrium problems. However, there are many scenarios where a single-period channel optimization solution may turn out to be myopic as it does not consider the after-effects of current pricing on future demand. To remedy this typical shortcoming, using general memory functions, we include the strategic customers’ cognitive bias toward pricing history in the supply channel equilibrium problem. In the form of two constructive theorems, we provide explicit solution algorithms for the ensuing Nash–Stackelberg equilibrium problems. In particular, we prove that our recursive solution algorithm can find equilibria in the multi-periodic variation of many standard supply channel contracts such as wholesale, buyback, and revenue-sharing contracts. Full article

The strategy method is often used in public goods games to measure an individual’s willingness to cooperate depending on the level of cooperation by their groupmates (conditional cooperation). However, while the strategy method is informative, it risks conflating confusion with a desire for fair outcomes, and its presentation may risk inducing elevated levels of conditional cooperation. This problem was highlighted by two previous studies which found that the strategy method could also detect equivalent levels of cooperation even among those grouped with computerized groupmates, indicative of confusion or irrational responses. However, these studies did not use large samples (n = 40 or 72) and only made participants complete the strategy method one time, with computerized groupmates, preventing within-participant comparisons. Here, in contrast, 845 participants completed the strategy method two times, once with human and once with computerized groupmates. Our research aims were twofold: (1) to check the robustness of previous results with a large sample under various presentation conditions; and (2) to use a within-participant design to categorize participants according to how they behaved across the two scenarios. Ideally, a clean and reliable measure of conditional cooperation would find participants conditionally cooperating with humans and not cooperating with computers. Worryingly, only 7% of participants met this criterion. Overall, 83% of participants cooperated with the computers, and the mean contributions towards computers were 89% as large as those towards humans. These results, robust to the various presentation and order effects, pose serious concerns for the measurement of social preferences and question the idea that human cooperation is motivated by a concern for equal outcomes. Full article

We investigate the strategic nature of farmers’ groundwater usage with a rich dataset from the American Midwest. We propose a new revealed preference test for the groundwater interaction as a dynamic game. We reject a view of groundwater usage decisions as strategic substitutes in favor of strategic complements as a better description. Full article

We test whether the decisions in a common-pool resource game are better modeled game-theoretically as strategic substitutes or complements using an individual-level dataset of groundwater usage that accounts for 3% of US irrigated agriculture. Based on a regression framework with instrumental variables, we find support for strategic complements, suggesting that reciprocity– and/or race-to-depletion–like dynamics are key to understanding groundwater usage. Full article

We consider a clearing problem in peer-to-peer energy markets, where prosumers can trade energy among each other and with the main grid to meet their energy demands. By using a game-theoretic formulation and exploiting operator-theoretic methods for generalized Nash equilibrium seeking, we propose two variants of the state-of-the-art distributed market clearing mechanism with improved convergence speeds. Furthermore, we design a third variant that allows for equilibrium selection, i.e., computing a specific market solution based on a convex preference function of the network operator, e.g., a congestion cost. We provide convergence guarantees and numerically show the advantages of our proposed algorithms in terms of convergence speed up and obtaining reduced grid congestion. Full article

We investigate whether tournament prizes that depend on joint output (“variable prize tournaments”) can alleviate the sabotage problem which is otherwise inherent in tournament structures. In a game-theoretical model with three contestants, we compare fixed-prize tournaments with tournaments where prizes depend on contestants’ joint output. Our analysis suggests that the incentives to sabotage in a fixed-prize tournament may be counteracted in a variable-prize tournament such that contestants no longer sabotage, but help one another. We empirically test the implications of our model with the help of a classroom experiment where we compare participants’ choices in a fixed-prize treatment ($FP$) with those of a variable-prize treatment ($VP$) in a between-subjects design. Given our parametrization, we expect efforts to be identical in both treatments, and we expect sabotage in the $FP$ treatment and no sabotage in the $VP$ treatment. In accordance with the model, we find that participants in the fixed-prize tournament sabotage one another, whereas participants in the variable-prize tournament help one another. At the same time, participants’ effort levels do not vary between the two treatments. Full article

We study two-sided matching contests with two sets, A and B, each of which includes a finite number of heterogeneous agents with commonly known types. The agents in each set compete in a lottery (Tullock) contest, and then are assortatively matched, namely, the winner of set A is matched with the winner of set B and so on until all the agents in the set with the smaller number of agents are matched. Each agent has a match value that depends on their own type and the type of their match. We assume that the agents’ efforts do not affect their match values and that they have a positive effect on welfare. Therefore, an interior equilibrium in which at least some of the agents are active is welfare superior to a corner equilibrium in which the agents choose to be non-active. We analyze the conditions under which there exists a (partial) interior equilibrium where at least some of the agents compete against each other and exert positive efforts. Full article

Why do some incomplete information markets feature intermediaries while others do not? I study the allocation of two goods in an incomplete information setting with a single principal, multiple agents with unit demand, and interdependent valuations. I construct a novel dynamic mechanism implemented by a principal who faces a set of intermediaries, each of whom represents an ex ante identical set of agents. This mechanism has a unique (up to permutation) weak perfect Bayesian equilibrium. The dynamic mechanism is inefficient with positive probability. Nevertheless, under mild conditions the agents are ex ante better off under the dynamic mechanism relative to a Vickrey-like auction because the intermediaries are more able to exploit information asymmetries in the dynamic mechanism than agents are able to exploit information asymmetries in the Vickrey-like auction. Finally, I show that in large markets the dynamic mechanism and Vickrey-like auction have the same expected total surplus. The comparison between the two mechanisms gives a stylized intuition for the hierarchical structure of larger markets and institutions. Full article

We study asymmetric second-price auctions under incomplete information. The bidders have two potentially different, commonly known, valuations for the object and private information about their entry costs. The seller, however, does not benefit from these entry costs. We calculate the equilibrium strategies of the bidders and analyze the optimal design for the seller in this environment in terms of expected entry and the number of potential bidders. Full article