Payoff Dominance Research Articles

Criminals Can Be Leaders: An Endogenous Timing Approach to Security Games

Security Games traditionally assume sequential interactions, with the defender acting as the leader and the attacker as the follower. In this paper, we challenge such an assumption by showing that security interactions can also be simultaneous or that even the attacker might be the leader. Specifically, we consider a simple security game framework consisting of two possible targets, one attacker and one defender. Then, we extend that basic interaction using a well-known endogenous timing framework (the Game with Observable Delay) to endogenously obtain the timing of the game as part of the equilibrium outcome. Our results show that, for the same set of general assumptions, the interaction can be simultaneous or sequential (with either the defender or attacker being the leader). Although the simultaneous case could be discarded using payoff dominance, it is impossible to compare the two sequential configurations using the same criterion. The risk dominance criterion also turns out to be helpless in this case. Thus, we argue that considering attackers as leaders in Security Games is a plausible scenario that merits further consideration, not only from a theoretical point of view but also due to its alignment with certain types of real-world crime.

Pure strategy Nash Equilibrium in 2-contestant generalized lottery Colonel Blotto games

We consider a large class of 2-contestant Colonel Blotto games, for which the budget and valuation are both asymmetric between players and the contest success functions are in Tullock form with battle-specific discriminatory power in (0,1] and battle-and-contestant-specific lobbying effectiveness. We prove the existence of pure strategy Nash Equilibrium of the game through a direct equilibrium characterization approach. We examine some important equilibrium properties. Among them, the rate of equilibrium rent dissipation ratio between the two contestants is constant across all the battles, and the payoff dominance relationship is impossible between any two distinct equilibria. Furthermore, we provide two sets of sufficient conditions, one for symmetric valuations and the other for asymmetric valuations, under each of which the equilibrium of the game is unique. Our study contributes to a better understanding regarding the existence, uniqueness, characterization and properties of equilibria in Blotto games.

Three-player game-theoretic allocation of indivisible resources during natural disasters

Resource allocation is an integral part of disaster management. After a natural disaster, multiple concurrent emergencies in distinct locations often make resource allocation challenging for the disaster management authority. This article focuses on disaster resource allocation based on a novel three-player, non-cooperative, single-stage, strategic game where the emergency locations are considered as the players. The decision-making authority solves a game-theoretic algorithm to determine a suitable strategy for allocating indivisible resources among three disaster locations where the available resources are insufficient to satisfy all the players simultaneously. Based on a unique non-monetary cost function, each player incurs a penalty for any possible allocation strategy. Mathematical analysis shows that, for the proposed game, at least one pure strategy Nash equilibrium (PSNE) always exists, which can be a desirable allocation to the players. Payoff dominance and utopia-point-based solutions are used to select a single PSNE from a set of multiple PSNEs.

A Method to Select Best Among Multi-Nash Equilibria

A non-zero sum bimatrix game may yield numerous Nash equilibrium solutions while solving the game. The selection of a good Nash equilibrium from among the many options poses a dilemma. In this article, three methods have been proposed to select a good Nash equilibrium. The first approach identifies the most payoff-dominant Nash equilibrium, while the second method selects the most risk-dominant Nash equilibrium. The third method combines risk dominance and payoff dominance by giving due weights to the two criteria. A sensitivity analysis is performed by changing the relative weights of criteria to check its effect on the ranks of the multiple Nash equilibria, infusing more confidence in deciding the best Nash equilibrium. JEL Codes: C7, C72, D81

Vertical differentiation, product innovation, and dynamic competition

AbstractThis paper presents an infinite horizon dynamic model in which two firms compete in a market vertically differentiated by the qualities of their products and consumers have heterogeneous preferences for quality. Given the product qualities offered, the firms engage in price competition that segments the market. In each period each firm can spend on product innovation that if successful increases the quality of its product. Three types of Markov perfect equilibria are identified. A running–coasting equilibrium exhibits increasing quality dominance with one firm undertaking innovation and the other coasting to free ride on the innovation by the first firm. The firm that coasts can have the larger dynamic payoff, so quality dominance does not imply payoff dominance. A second is a leap‐frog equilibrium in which the trailing firm undertakes innovation to leap into the lead. The trailing firm never innovates solely to narrow the gap with the leader, so catch up strategies are never used. In the third both firms undertake innovation, but if both have innovation successes, product differentiation remains the same and profits are reduced by the cost of innovation. The rivalry between Intel and AMD in microprocessors for personal computers provides a motivating example.

Nash versus coarse correlation

We run a laboratory experiment to test the concept of coarse correlated equilibrium (Moulin and Vial in Int J Game Theory 7:201–221, 1978), with a two-person game with unique pure Nash equilibrium which is also the solution of iterative elimination of strictly dominated strategies. The subjects are asked to commit to a device that randomly picks one of three symmetric outcomes (including the Nash point) with higher ex-ante expected payoff than the Nash equilibrium payoff. We find that the subjects do not accept this lottery (which is a coarse correlated equilibrium); instead, they choose to play the game and coordinate on the Nash equilibrium. However, given an individual choice between a lottery with equal probabilities of the same outcomes and the sure payoff as in the Nash point, the lottery is chosen by the subjects. This result is robust against a few variations. We explain our result as selecting risk-dominance over payoff dominance in equilibrium.

Conditional behavior and learning in similar stag hunt games

This paper reports an experiment using Stag Hunt games with different payoff ranges. Specifically, in one treatment for half of the games the payoff dominant and the risk dominant equilibrium coincide and for the other half of the games they conflict; in the second treatment they always conflict. The experiment provides evidence that the payoff range experienced by the participant influences the likelihood of efficient conventions emerging. In particular, experiencing games where payoff dominance and risk dominance coincide appears to make payoff dominance more attractive in games in which they conflict. In the experiment, we also observe conditional behavior emerging with experience. We develop a model of conditional expectations to explain these stylized facts that depends crucially on the assumption that after a brief learning period participants categorize their experience using the same relative bandwidth in both treatments even though the range of experience is twice as large in treatment 1 as it is in treatment 2. The assumption cannot be rejected by the data. The analysis provides a formal example in which increasing experienced diversity by changing the way similar experiences are categorized increases the likelihood of efficient conventions emerging in communities playing similar Stag Hunt games.

When less information is good enough: experiments with global stag hunt games

There is mixed evidence on whether subjects coordinate on the efficient equilibrium in experimental stag hunt games under complete information. A design that generates an anomalously high level of coordination, Rankin et al. (Games Econo Behav 32(2):315–337, 2000), varies payoffs each period in repeated play rather than holding them constant. These payoff “perturbations” are eerily similar to those used to motivate the theory of global games, except the theory operates under incomplete information. Interestingly, that equilibrium selection concept is known to coincide with risk dominance, rather than payoff dominance. Thus, in theory, a small change in experimental design should produce a different equilibrium outcome. We examine this prediction in two treatments. In one, we use public signals to match Rankin et al. (2000)’s design; in the other, we use private signals to match the canonical example of global games theory. We find little difference between treatments, in both cases, subject play approaches payoff dominance. Our literature review reveals this result may have more to do with the idiosyncrasies of our complete information framework than the superiority of payoff dominance as an equilibrium selection principle.

Impulse Response Dynamics in Weakest Link Games

Abstract In a recent paper, Croson et al. (2015) experimentally study three weakest link games with multiple symmetric equilibria. They demonstrate that static concepts based on the Nash equilibrium (including multiple Nash equilibria, quantal response equilibria, and equilibrium selection by risk and payoff dominance) cannot successfully capture the observed treatment differences. Using Reinhard Selten’s impulse response dynamics, we derive a proposition that provides a theoretical ranking of contribution levels in the weakest link games. We show that the predicted ranking of treatment outcomes is fully consistent with the observed data. In addition, we demonstrate that the impulse response dynamics perform well in tracking average contributions over time. We conclude that Reinhard Selten’s impulse response dynamics provide an extremely valuable behavioral approach that is not only capable of resolving the indecisiveness of static approaches in games with many equilibria, but that can also be used to track the development of choices over time in games with repeated interaction.

Using Modified UCT Algorithm Basing on Risk Estimation Methods in Imperfect Information Games

Risk dominance and payoff dominance strategy are two complementary parts of the game theory decision strategy. While payoff dominance is still the basic principle in perfect information, two player games, risk dominance has shown its advantages in imperfect information conditions. In this paper, we first review the related work in the area of estimation methods and the influence of risk factors on computing game equilibrium. Then a new algorithm, UCT-Risk is proposed in this paper, which is a modification of UCT (UCB apply to Trees) algorithm based on risk estimation methods. Finally, we implement the proposed algorithm in SiGuo game, a popular imperfect information game in China. The experimental result of the new algorithm shows it correctness and effectiveness.

Modified UCT Algorithm with Risk Dominance Methods in Imperfect Information Game

UCT (Upper confidential bounds on Trees) has been applied quite well as a selection approach in MCTS(Monte Carlo Tree Search) in imperfect information games like poker. By using risk dominance as complementary part of decision method besides payoff dominance, opponent strategies is better characterized as their risk factors, like bluff actions in Texas Hold’em Poker . In this paper, estimation method about the influence of risk factors on computing game equilibrium is provided. A novel algorithm, UCT-risk is proposed as modification about UCT algorithm basing on risk estimation methods. To verify the performance of new algorithm, Texas Hold’em, a popular test-bed for AI research is chosen as the experiment platform. The Agent adopted UCT-risk algorithm performs as well or better as the best previous approaches in experiments. And also it is applied in a poker agent named HITSZ_CS_13 in the 2013 AAAI Computer Poker Competition, which confirms the effectiveness of the UCT-risk provided in this paper.

Strategic Treaty Shopping

Treaty shopping refers to a situation in which an individual or a company that is not eligible for the benefits of a tax treaty uses an intermediary entity that is eligible for such benefits to obtain these benefits indirectly. This paper examines treaty shopping with a game-theoretic model in a network of tax treaties. An investor can choose an investment route across national borders to minimize tax while a tax agency can choose to audit the investor to find out the route. The audit is costly but it can give additional revenue to the tax agency if it reveals that the investor chose an indirect route for tax avoidance. I analyze the equilibrium of this model, calculate tax revenue loss due to treaty shopping, and show the payoff dominance of a random audit strategy to the always-audit strategy. I also examine the structure of tax-minimizing investment routes in a real-world network of tax treaties between selected countries. While about 76 percent of tax-minimizing indirect routes pass through countries with no withholding tax, about 21 percent of tax-minimizing indirect routes pass through countries with tax treaty networks favorable to certain residence countries. Network centrality measures are introduced to assess the role of pass-through countries. To prevent treaty shopping countries may amend tax treaties with possible pass-through countries.

THE DYNAMICS OF NORMS AND CONVENTIONS UNDER LOCAL INTERACTIONS AND IMITATION

Restricting the analysis to general 2×2 coordination games, this article shows how under certain conditions, it is highly likely that individuals coordinate on a (pay-off) efficient through risk inferior convention. This contrasts with other equilibrium refinement criteria, such as risk dominance or stochastic stability. Here it is assumed that players are situated on a toroidal regular lattice, interact only locally and, in each period, imitate the last period's most successful player in their neighborhood. If the set of observable players by an individual and the set that he interacts with are both identical and small, pay-off dominance plays the major role in defining the long-term convention. As the latter set of players increases, a risk dominant but pay-off inferior convention becomes more likely. The model also shows that the interaction of two player types in a nonsymmetric game potentially leads to nonegalitarian conventions.

Cognitive Resource Depletion, Choice Consistency, and Risk Preferences

We investigate the consistency and stability of individual risk preferences by slightly manipulating the cognitive resources of subjects through sleepiness. Participants are recruited and randomly assigned to an experiment session at a preferred time of day relative to their diurnal preference (circadian matched) or at a non-preferred time of day (circadian mismatched). For the decision task, subjects and are asked to choose how much to allocate between two state-dependent assets (using the Choi et al., 2007 design). We have two main findings. First, the consistency of behavior for circadian matched and mismatched subjects is statistically the same. This is true whether it is (non-parametrically) defined as consistency with GARP, payoff dominance, expected utility, disappointment aversion, or cumulative prospect theory. Second, while our cognitive resource manipulation yields no difference in consistency of behavior, it results in an increased tendency to take risk. Our experiment confirms theoretical predictions that preferences are consistent yet state-dependent.

Downlink Power Control for CDMA Satellite Cognitive Radio

In this paper, we consider power control for a code-division-multiple-access (CDMA)-based satellite cogni- tive radio network where the satellite acts as the common transmitter of primary users and secondary users. Specially, we assume that primary users are willing to share resource with secondary users by means of leasing spectrum, while secondary users have to pay for the shared spectrum. In addition, secondary users are allowed to transmit on any sub-channel provided that the resulting interference to any primary users is below a critical threshold. We focus on the downlink. The objective is to maximize the throughput in the licensed frequency band of the satellite network. We formulate the problem as a game theoretic problem with all users as the players. We propose a tow-phase control scheme that can be accepted willingly by both types of users and can maximize the throughput of the satellite network. In the first phase, the Nash Equilibrium point is calculated with an iterative method, the existence of the Nash Equilibrium point is also proved. In the second phase, Payoff Dominance Selection is used to choose the optimal power allocation under the constraints including quality of service protections of all users and maximum transmission power of the onboard power amplifier. Simulation is performed to study the parameters of the system and of each user. By simulation, we find that the throughput is greatly increased by the proposed scheme compared with traditional satellite networks where fixed power is allocated to only primary users. Onboard complexity is also analyzed.

Technology adoption in markets with network effects: Theory and experimental evidence

We examine a technology-adoption game with network effects in which coordination on either technology A or technology B constitutes a Nash equilibrium. Coordination on technology B is assumed to be payoff dominant. We define a technology’s critical mass as the minimal share of users, which is necessary to make the choice of this technology the best response for any remaining user. We show that the technology with the lower critical mass implies risk dominance and selection by the maximin criterion. We present experimental evidence that both payoff dominance and risk dominance explain participants’ choices in the technology-adoption game. The relative riskiness of a technology can be proxied using either technologies’ critical masses or stand-alone values absent any network effects.

Mutual support in games: Some properties of Berge equilibria

The Berge equilibrium concept formalizes mutual support among players motivated by the altruistic social value orientation in games. We prove some basic results for Berge equilibria and their relations to Nash equilibria, and we provide a straightforward method for finding Berge equilibria in n -player games. We explore some specific examples, and we explain how the Berge equilibrium provides a compelling model of cooperation in social dilemmas. We show that the Berge equilibrium also explains coordination in some common interest games and is partially successful in explaining the payoff dominance phenomenon, and we comment that the theory of team reasoning provides alternative solutions to these problems.

Coordination Games and Local Interactions: A Survey of the Game Theoretic Literature

We survey the recent literature on coordination games, where there is a conflictbetween risk dominance and payoff dominance. Our main focus is on models of local interactions, where players only interact with small subsets of the overall population rather than with society as a whole. We use Ellison’s [1] Radius-Coradius Theorem to present prominent results on local interactions. Amongst others, we discuss best reply learning in a global- and in a local- interaction framework and best reply learning in multiple location models and in a network formation context. Further, we discuss imitation learning in a localandin a global- interactions setting.

Payoff dominance and risk dominance in the observable delay game: a note

We examine whether the payoff dominant sequential-move (Stackelberg) outcome is realized when timing is endogenized. We adopt the observable delay game formulated by Hamilton and Slutsky [Games Econ Behav 2(1):29–46, 1990]. We find that if one sequential-move outcome is payoff dominant, either (i) the outcome both players prefer is the unique equilibrium; or (ii) two sequential-move outcomes are equilibria and the one both players prefer is risk dominant. In other words, no conflict between payoff dominance and risk dominance in the observable delay game exists, in contrast to other games such as (non pure) coordination games. We also find that even if one of two sequential-move outcomes is the unique equilibrium outcome in the observable delay game, it does not imply that the equilibrium outcome is payoff dominant to the other sequential-move outcome.

Near-Efficient Equilibria in Collaborative Meritocracies

We examine theoretically and experimentally how a society's grouping and stratification rules affect incentives and efficiency, and compare meritocratic and ascriptive grouping. We present a multi-level model of social production, and extend the usual single-group-level analysis of cooperation to a broadly defined system, in which individuals compete for inclusion in stratified groups based on the contributions they make. Group members share their collective output equally amongst themselves. The mechanism has two pure strategy Nash equilibria, one close to Pareto optimal. The latter equilibrium is asymmetric and rather complex for experimental subjects. Nonetheless, subjects tacitly coordinate this equilibrium very reliably, demonstrating equilibrium's predictive power and providing empirical support for payoff dominance. Our behavioral findings also point to a meritocracy's naturalness and effectiveness in eliciting high social contributions. The results make a theoretical and empirical case for why social organization should be based on contribution rather than privilege. They also indicate why societies are increasingly becoming performance-based meritocracies, and are relevant to many forms of contemporary social organization.