Symmetric Games Research Articles

Does moral play equilibrate?

Some finite and symmetric two-player games have no (pure or mixed) symmetric Nash equilibrium when played by partly morally motivated players.The reason is that the “right thing to do” may be not to randomize. We analyze this issue both under complete information between equally moral players and under incomplete information between players of arbitrary degrees of morality. We provide necessary and sufficient conditions for the existence of equilibrium and illustrate the results with examples and counter examples.

Learning by similarity-weighted imitation in winner-takes-all games

We study a simple model of similarity-based global cumulative imitation in symmetric games with large and ordered strategy sets and a salient winning player. We show that the learning model explains behavior well in both field and laboratory data from one such “winner-takes-all” game: the lowest unique positive integer game in which the player that chose the lowest number not chosen by anyone else wins a fixed prize. We corroborate this finding in three other winner-takes-all games and discuss under what conditions the model may be applicable beyond this class of games. Theoretically, we show that global cumulative imitation without similarity weighting results in a version of the replicator dynamic in winner-takes-all games.

Sion's minimax theorem and Nash equilibrium of symmetric three-players zero-sum game

About a symmetric three-players zero-sum game we will show the following results. 1. A modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy are proved by the existence of a symmetric Nash equilibrium. 2. The existence of a symmetric Nash equilibrium is proved by the modified version of Sion's minimax theorem with the coincidence of the maximin strategy and the minimax strategy. Thus, they are equivalent. If a zero-sum game is asymmetric, maximin strategies and minimax strategies of players do not correspond to Nash equilibrium strategies. If it is symmetric, the maximin strategies and the minimax strategies constitute a Nash equilibrium. However, without the coincidence of the maximin strategy and the minimax strategy there may exist an asymmetric equilibrium in a symmetric three-players zero-sum game.

A Genetic Algorithm approach to Asymmetrical Blotto Games with Heterogeneous Valuations

Blotto Games are a popular model of multi-dimensional strategic resource allocation. Two players allocate resources in different battlefields in an auction setting. While competition with equal budgets is well understood, little is known about strategic behavior under asymmetry of resources, as this situation is hardly solvable with analytical methods. We introduce a genetic algorithm, a search heuristic inspired from biological evolution, interpreted as a mechanism of social learning, to solve this problem. Through multi-agent reinforcement learning, the algorithm evaluates the performance of random initial strategies. Most performant strategies are combined to create more performant strategies. Mutations allow the algorithm to efficiently scan the space of possible strategies, and consider a wide diversity of deviations. Iterating this process improves strategies until no profitable deviation exists. This method allows to identify optimal solutions to problems involving very large strategy spaces, such as Blotto games. We show that our genetic algorithm converges to the analytical Nash equilibrium of the symmetric Blotto game. We present the solution concept it provides for asymmetrical Blotto games. It notably sees the emergence of ``guerilla warfare'' strategies, consistent with empirical and experimental findings. The player with less resources learns to concentrate its resources to compensate for the asymmetry of competition. When players value battlefields heterogeneously, counter strategies and bidding focus is obtained in equilibrium. These features are consistent with empirical and experimental findings, and provide a learning foundation for their existence. These results open towards a more complete characterization of solutions in strategic resource allocation games. It places genetic algorithms as a search method of interest in game theoretical problems, to achieve further theoretical and operational achievements.

The Two-Stage Game Approach to Coalition Formation: Where We Stand and Ways to Go

Coalition formation is often analysed in an almost non-cooperative way, as a two-stage game that consists of a first stage comprising membership actions and a second stage with physical actions, such as the provision of a public good. We formalised this widely used approach for the case where actions are simultaneous in each stage. Herein, we give special attention to the case of a symmetric physical game. Various theoretical results, in particular, for cartel games, are provided. As they are crucial, recent results on the uniqueness of coalitional equilibria of Cournot-like physical games are reconsidered. Various concrete examples are included. Finally, we discuss research strategies to obtain results about equilibrium coalition structures with abstract physical games in terms of qualitative properties of their primitives.

Evolution and Heterogeneity of Social Preferences

Why do individuals take different decisions when confronted with similar choices? This paper investigates whether the answer lies in an evolutionary process. Our analysis builds on recent work in evolutionary game theory showing the superiority of a given type of preferences, homo moralis, in fitness games with assortative matching. We adapt the classical definition of evolutionary stability to the case where individuals with distinct preferences coexist in a population. This approach allows us to establish the characteristics of an evolutionarily stable population. Then, introducing an assortment matrix for assortatively matched interactions, we prove the existence of a heterogeneous evolutionarily stable population in 2×2 symmetric fitness games under constant assortment, and we identify the conditions for its existence. Conversely to the classical setting, we find that the favored preferences in a heterogeneous evolutionarily stable population are context-dependent. As an illustration, we discuss when and how an evolutionarily stable population made of both selfish and moral individuals exists in a prisoner’s dilemma. These findings offer a theoretical foundation for the empirically observed diversity of preferences among individuals.

The Determinants of Efficient Behavior in Coordination Games

We study the determinants of efficient behavior in stag hunt games (2x2 symmetric coordination games with Pareto ranked equilibria) using both data from eight previous experiments on stag hunt games and data from a new experiment which allows for a more systematic variation of parameters. In line with the previous experimental literature, we find that subjects do not necessarily play the efficient action (stag). While the rate of stag is greater when stag is risk dominant, we find that the equilibrium selection criterion of risk dominance is neither a necessary nor sufficient condition for a majority of subjects to choose the efficient action. We do find that an increase in the size of the basin of attraction of stag results in an increase in efficient behavior. We show that the results are robust to controlling for risk preferences.

Information Design in Coordination Games with Risk Dominant Equilibrium Selection

We study the design of public information structures that maximize the probability of selecting a Pareto dominant equilibrium in symmetric (2 x 2) coordination games. Because the need to coordinate exposes players to strategic risk, we treat the designer as able to implement an equilibrium only if the players believe it is also risk dominant. The designer's task is therefore to pool the set of states in which the desired equilibrium is risk dominant with the largest possible set in which it is not, while keeping the desired equilibrium risk dominant in expectation. We provide a simple characterization of the optimal signal structure which holds under general conditions. We extend the analysis to related problems, and show that our intuition is robust, suggesting that our approach provides a promising way forward for a large class of problems in constrained information design.

Robustness Against Indirect Invasions in Asymmetric Games

The concept of robustness against indirect invasions is well-known for symmetric games. We are concerned with the technical aspects and relevance of this concept for asymmetric games with continuous strategy space. For such games, we show that the set of all indirect neutral mutants of a robust profile is equivalent to a minimal evolutionarily stable set. It is also proved that a globally strong uninvadable profile is robust and the set of its indirect neutral mutants is a singleton. The results are illustrated using examples.

Greed and Fear in Downstream R&D Games

The aim of this paper is to investigate the fi rms’ incentives to engage in process R&D under vertical industrial setting, when the raising rivals’ cost effect is present. We show that R&D investment of the downstream duopoly fi rm raises the rival’s marginal costs of production. The downstream R&D behavior can give rise to the symmetric investment games, i.e., the prisoner’s dilemma, the deadlock game and the harmony game, between downstream competitors. If the costs of the R&D investments made by the downstream fi rms are large enough, the downstream fi rms can participate in the harmony game, which results in the investment hold-up or the creation of the R&D-avoiding cartel. For more R&D-effi cient downstream fi rms, the downstream investment game can end up in the prisoner’s dilemma or the deadlock game. In the prisoner’s dilemma, both downstream fi rms invest in R&D, but such a behavior is not Pareto optimal. In the prisoner’s dilemma, greed and fear make fi rms invest in R&D. In the deadlock game, both downstream fi rms invest in R&D, and such a behavior is Pareto optimal. The R&D investments are not induced by any social tension (greed or fear).

Capacity choice game in a multiserver queue: Existence of a Nash equilibrium

AbstractIn many congestion‐prone services, front‐line employees have discretion over the rate at which they serve customers. To evaluate the impact of queue pooling on their decisions, we model the situation as a two‐server, single‐queue symmetric capacity choice game. Gopalakrishnan et al. (2016) characterize the existence of a Nash equilibrium in this game under a requirement on the servers' capacity cost functions, that is, where servers have limited discretion. Without that requirement and when servers are free to choose any service rate, the servers' cost function is ill‐behaved and standard tools for establishing the existence of an equilibrium cannot be applied. We consider a general power capacity cost function with no restriction on the servers' choice of capacity, and rely on a lesser‐known result, namely Tarski's intersection theorem, to establish the existence of a symmetric pure‐strategy Nash equilibrium. Comparing settings where queue stability is enforceable versus not, we show that there always exists a Nash equilibrium in the former case, unlike in the latter, and that some of the capacity choices that are equilibria in the former case are no longer equilibria in the latter. Our analysis highlights the criticality of the enforceability of system stability on equilibrium outcomes.

Pareto-optimal solution for the quantum battle of the sexes

Quantum games have gained much popularity in the last two decades. Many of these quantum games are a redefinition of iconic classical games to fit the quantum world, and they gain many different properties and solutions in this different view. In this letter, we attempt to find a solution to an asymmetric quantum game which still troubles quantum game researchers, the quantum battle of the sexes. To achieve that, we perform an analysis using the Eisert-Wilkens-Lewenstein's protocol for this asymmetric game. The protocol highlights two solutions for the game, which solve the dilemma and satisfy the Pareto-optimal definition, unlike previous reports that rely on Nash equilibrium. We perform an experimental implementation using the NMR technique in a two-qubit system. Our results eliminate dilemmas on the quantum battle of the sexes and provide us with arguments to elucidate that the Eisert-Wilkens-Lewenstein's protocol is not restricted to symmetric games when at the quantum regime.

Evolutionary Game Theory: A Generalization of the ESS Definition

In this paper, we study the concept of Evolutionarily Stable Strategies (ESSs) for symmetric games with [Formula: see text] players. The main properties of these games and strategies are analyzed and several examples are provided. We relate the concept of ESS with previous literature and provide a proof of finiteness of ESS in the context of symmetric games with [Formula: see text] players. We show that unlike the case of [Formula: see text], when there are more than two populations an ESS does not have a uniform invasion barrier, or equivalently, it is not equivalent to the strategy performing better against all strategies in a neighborhood. We also construct the extended replicator dynamics for these games and we study an application to a model of strategic planning of investment.

Volunteers Dilemma Game Inspired Broadcast Scheme for Vehicular Ad Hoc Networks

Multi-hop broadcast communication is one of the main schemes for supporting message dissemination in vehicular ad hoc networks. In this paper, we propose an efficient receiver-oriented broadcast scheme in which the receiving vehicles' probability of forwarding is modeled by the symmetric volunteer's dilemma game. Based on this game, the vehicles that receive the broadcast message are players. At least one of the players should pay a cost and be a volunteer to rebroadcast the message, and then all will benefit from this volunteering. Utilizing fuzzy logic techniques and considering information from the network layer about local density and probability of transmission, the contention window size at the MAC layer will be adjusted. We develop ns-3 simulations to evaluate the performance of the proposed volunteer's dilemma inspired broadcast (VDIB) scheme in terms of reachability, number of rebroadcasts per covered vehicle, number of bytes sent per covered vehicle, and per-hop delay. VDIB performance is compared with that of the distance-to-mean broadcast method, broadcast component of contention-based forwarding, distribution-adaptive distance with channel quality, statistical location assisted broadcast, fuzzy logic-based broadcast, bandwidth efficient fuzzy logic assisted broadcast, and intelligent hybrid adaptive broadcast protocols in both highway and urban environments.

Cyber Technology and the Arms Race

Abstract Cyber technology represents digital military capability with the purpose of causing damage to the military strength of a potential enemy. War using conventional weapons may be preceded by a strike using cyber technology. This paper introduces such technology into the theory of conflicts. The cost of war relative to the payoff from victory turns out to be crucial for the results on armament decisions. In the war game, two types of Nash equilibria may arise. One is subject to warfare while the other is not (‘equilibrium of terror’), depending on the perceived cost of war. In a symmetric war game, cyber capabilities are neutral with respect to the investments in conventional weapons, but they make wars more likely. Asymmetric access to cyber technology limits the international arms race with conventional weapons. A low success probability in the cyber programme encourages exercising the cyberattack option as the enemy may not have access to cyber capability. Uncertainty of the success of a cyber programme makes countries cautious when allocating resources not only to these programmes but also in conventional armament.

Linear-State Differential Games in Partition Function Form

We introduce a partition function for [Formula: see text]-player linear-state cooperative differential games. The value of a coalition within a given coalition structure is defined as its noncooperative equilibrium payoff of a game played between the coalitions. We also define two core notions, namely, the cautious and the singleton core. If the game is convex, then the cores are nonempty. In order to illustrate the approach, we consider a symmetric game of pollution accumulation.

Zero-determinant strategies in repeated asymmetric games

Zero-determinant (ZD) strategies are conditional strategies which allow players adopting them to establish a relation between their expected payoffs and those of their opponents. The ZD strategies were investigated in different models including finite and infinite repeated two players’ games, multiplayer games, continuous action spaces and alternating move games. However, all previous studies considered only symmetric games between players, i.e., players have the same strategies and the same associated payoffs, thus the players’ identities are interchangeable without affecting the game's dynamics. In this study, we analytically model and derive the ZD strategies for asymmetric two players’ games, focusing on one-memory strategies and infinite repeated encounters. We derive the analytical bounds of equalizer and extortionate ZD strategies in 2 × 2 asymmetric games, which differ from the symmetric games case. Furthermore, we derive under what conditions a player using an extortionate ZD strategy will get a higher expected payoff than his/her opponent. Finally using a numerical example, we investigate ZD strategies in 2 × 2 asymmetric prisoner's dilemma game.

An iterative algorithm based on strong [formula omitted]-tensors for identifying positive definiteness of irreducible homogeneous polynomial forms

Identifying the positive definiteness of an even-degree homogeneous polynomial form arises in various applications, such as evolutionary game dynamics, automatical control, magnetic resonance imaging, and spectral hypergraph theory. However, it is difficult to determine whether a given homogeneous polynomial is positive definite or not because the problem is NP-hard. In this paper, an iterative algorithm of identifying the positive definiteness of irreducible homogeneous polynomial forms is proposed by identifying strong H-tensors with weakly irreducibility. The validity of the iterative algorithm is proved theoretically. Experimental results on multilinear systems, high-order Markov chains and symmetric multi-player games are presented to illustrate the applications of the proposed methods.

The Inefficiency of Nash and Subgame Perfect Equilibria for Network Routing

This paper provides new bounds on the quality of equilibria in finite congestion games with affine cost functions, specifically for atomic network routing games. It is well known that the price of anarchy equals exactly 5/2 in general. For symmetric network routing games, it is at most (5n − 2)/(2n + 1), where n is the number of players. This paper answers to two open questions for congestion games. First, we show that the price of anarchy bound (5n − 2)/(2n + 1) is tight for symmetric network routing games, thereby answering a decade-old open question. Second, we ask whether sequential play and subgame perfection allows to evade worst-case Nash equilibria, and thereby reduces the price of anarchy. This is motivated by positive results for congestion games with a small number of players, as well as recent results for other resource allocation problems. Our main result is the perhaps surprising proof that subgame perfect equilibria of sequential symmetric network routing games with linear cost functions can have an unbounded price of anarchy. We complete the picture by analyzing the case with two players: we show that the sequential price of anarchy equals 7/5 and that computing the outcome of a subgame perfect equilibrium is NP-hard.

Game theory of vaccination and depopulation for managing livestock diseases and zoonoses on small-scale farms

Livestock producers adapt their farm management to epidemiological risks in different ways, through veterinary interventions but also by modulating their farm size and the removal rate of animals. The objective of this theoretical study was to elucidate how these behavioral adaptations may affect the epidemiology of highly-pathogenic avian influenza in domestic poultry and the outcome of the implemented control policies. We studied a symmetric population game where the players are broiler poultry farmers at risk of infection and where the between-farms disease transmission is both environmental and mediated by poultry trade. Three types of farmer behaviors were modelled: vaccination, depopulation, and cessation of poultry farming. We found that the transmission level of the disease through trade networks has strong qualitative effects on the system's epidemiological-economic equilibria. In the case of low trade-based transmission, when the monetary cost of infection is high, depopulation behavior can maintain a stable disease-free equilibrium. In addition, vaccination behavior can lead to eradication by private incentives alone – an outcome not seen for human diseases. In a scenario of high trade-based transmission, depopulation behavior has perverse epidemiological effects as it accelerates the spread of disease via poultry trade. In this situation, state interventions should focus on making vaccination technologies available at a low price rather than penalizing infected farms.