Mixed Nash Equilibrium Research Articles

What if criminals optimize their choice? Optimal strategies to defend public goods

We investigate optimal strategies to defend valuable goods against the attacks of a thief. Given the value of the goods and the probability of success for the thief, we look for the strategy that assures the largest benefit to each player irrespective of the strategy of his opponent. Two complementary approaches are used: agent-based modeling and game theory. It is shown that the compromise between the value of the goods and the probability of success defines the mixed Nash equilibrium of the game, that is compared with the results of the agent-based simulations and discussed in terms of the system parameters.

Implementation in mixed Nash equilibrium

A mechanism implements a social choice correspondence f in mixed Nash equilibrium if, at any preference profile, the set of all (pure and mixed) Nash equilibrium outcomes coincides with the set of f-optimal alternatives for all cardinal representations of the preference profile. Unlike Maskinʼs definition, our definition does not require each optimal alternative to be the outcome of a pure equilibrium. We show that set-monotonicity, a weakening of Maskinʼs monotonicity, is necessary for mixed Nash implementation. With at least three players, set-monotonicity and no-veto power are sufficient. Important correspondences that are not Maskin monotonic can be implemented in mixed Nash equilibrium.

Spike-based Decision Learning of Nash Equilibria in Two-Player Games

Humans and animals face decision tasks in an uncertain multi-agent environment where an agent's strategy may change in time due to the co-adaptation of others strategies. The neuronal substrate and the computational algorithms underlying such adaptive decision making, however, is largely unknown. We propose a population coding model of spiking neurons with a policy gradient procedure that successfully acquires optimal strategies for classical game-theoretical tasks. The suggested population reinforcement learning reproduces data from human behavioral experiments for the blackjack and the inspector game. It performs optimally according to a pure (deterministic) and mixed (stochastic) Nash equilibrium, respectively. In contrast, temporal-difference(TD)-learning, covariance-learning, and basic reinforcement learning fail to perform optimally for the stochastic strategy. Spike-based population reinforcement learning, shown to follow the stochastic reward gradient, is therefore a viable candidate to explain automated decision learning of a Nash equilibrium in two-player games.

N-Player Quantum Games in an EPR Setting

The -player quantum games are analyzed that use an Einstein-Podolsky-Rosen (EPR) experiment, as the underlying physical setup. In this setup, a player’s strategies are not unitary transformations as in alternate quantum game-theoretic frameworks, but a classical choice between two directions along which spin or polarization measurements are made. The players’ strategies thus remain identical to their strategies in the mixed-strategy version of the classical game. In the EPR setting the quantum game reduces itself to the corresponding classical game when the shared quantum state reaches zero entanglement. We find the relations for the probability distribution for -qubit GHZ and W-type states, subject to general measurement directions, from which the expressions for the players’ payoffs and mixed Nash equilibrium are determined. Players’ payoff matrices are then defined using linear functions so that common two-player games can be easily extended to the -player case and permit analytic expressions for the Nash equilibrium. As a specific example, we solve the Prisoners’ Dilemma game for general . We find a new property for the game that for an even number of players the payoffs at the Nash equilibrium are equal, whereas for an odd number of players the cooperating players receive higher payoffs. By dispensing with the standard unitary transformations on state vectors in Hilbert space and using instead rotors and multivectors, based on Clifford’s geometric algebra (GA), it is shown how the N-player case becomes tractable. The new mathematical approach presented here has wide implications in the areas of quantum information and quantum complexity, as it opens up a powerful way to tractably analyze N-partite qubit interactions.

Multi-Stakeholder Road Pricing Game: Solution Concepts

A road pricing game is a game where various stakeholders and/or regions with different (and usually conflicting) objectives compete for toll setting in a given transportation network to satisfy their individual objectives. We investigate some classical game theoretical solution concepts for the road pricing game. We establish results for the road pricing game so that stakeholders and/or regions playing such a game will beforehand know what is obtainable. This will save time and argument, and above all, get rid of the feelings of unfairness among the competing actors and road users. Among the classical solution concepts we investigate is Nash equilibrium. In particular, we show that no pure Nash equilibrium exists among the actors, and further illustrate that even “mixed Nash equilibrium” may not be achievable in the road pricing game. The paper also demonstrates the type of coalitions that are not only reachable, but also stable and profitable for the actors involved.

Second container port in Sri Lanka; Hambanthota or Trincomalee: an analysis using the game theory

Despite its early start, Colombo remains being the only containerised port of Sri Lanka. A second container port, therefore, would be desirable in solving the prevailing capacity problem. Hambanthota and Trincomalee appear to be the ideal candidates for this position. For this study, the overall expected utility at the mixed Nash Equilibrium was calculated for Hambanthota and Trincomalee, analysing opinion survey data to choose the viable outcome with better pay-off from the given two alternatives. The results indicate that the level of preference for Trincomalee as the second container port is significantly higher than that for Hambanthota.

Separating the Hawks from the Doves: Evidence from continuous time laboratory games

Human players in our laboratory experiment received flow payoffs over 120 seconds each period from a standard Hawk–Dove bimatrix game played in continuous time. Play converged closely to the symmetric mixed Nash equilibrium under a one-population matching protocol. When the same players were matched in a two-population protocol with the same bimatrix, they showed clear movement towards an asymmetric (and very inequitable) pure Nash equilibrium of the game. These findings support distinctive predictions of evolutionary game theory.

The Experimental Research on 2×2 Game with Unique Mixed Strategy Nash Equilibrium

This paper reports the results of an experimental study of playing a simple 2×2 non-zero sum game with asymmetrical payoff function. Subjects make decisions about their estimation for their opponent, and then each of them chooses a strategy after their roles are changed at random. As a result, steady states which subjects estimates are quite different from those predicted by the Nash equilibrium theory. Through comparative analysis, it is found that the selections of strategies are related to others’ payoff as well as subjects’ own return. And what is more, the subjects’ risk attitudes influence the choice of game.

Social Transition Spectrum in Constant Sum 2x2 Games with Human Subjects

In game, with the individual interactions, the social state is changing stochastically and governed by Nash Equilibrium in the strategy space. With the method from the physics of nonequlibrium steady state (NESS) and the data from experimental economics treatments (EET), we obtain the social transition spectrums and from which, the kink phenomena and the fluctuation theorem predictions in NESS are supported. The data comes from the fi xed paired 2-person constant sum 2x2 games with unique mixed strategy Nash equilibrium (MSNE). These findings demonstrate the new experimetric for social dynamics, meanwhile, an indiff erence within social and physical systems.

An Evolutionary Approach in Financial Forecasting

We develop an adaptive learning game to rethink efficient markets. We use the stochastically stable state of this game, which is a mixed Nash equilibrium, to form an adaptive expectation model that provides an estimate of the confidence interval for prices on the next day. The estimate is most accurate in the time of bubbles and crises, when rational expectations no longer fully hold.

How do People Play Against Nash Opponents?

We examine experimentally how humans behave when they play against a computer which implements its part of a mixed strategy Nash equilibrium. We consider two games, one zero-sum and another unprofitable with a pure minimax strategy. A minority of subjects' play was consistent with their Nash equilibrium strategy, while a larger percentage of subjects' play was more consistent with different models of play: equiprobable play for the zero-sum game, and the minimax strategy in the unprofitable game. We estimate the heterogeneity and dynamics of the subjects' latent mixed strategy sequences via a hidden Markov model. This provides clear results on the identification of the use of pure and mixed strategies and the limiting distribution over strategies. The mixed strategy Nash equilibrium is not self-enforcing except when it coincides with the equal probability mixed strategy, and there is surprising amounts of pure strategy play and clear cycling between the pure strategy states.

The Fashion Game and the Fashion Curse

It is impossible, in general, to extend an asymmetric two-player game to networks, because there must be two populations, the row one and the column one, but we do not know how to define inner-population interactions. This is not the case for Matching Pennies, as we can interpret the row player as a conformist, who prefers to coordinate her opponent's action, while the column player can be interpreted as a rebel, who likes to anti-coordinate. Therefore we can naturally define the interaction between two conformists as the coordination game, and that between two rebels as the anti-coordination game. It turns out that the above network extension of Matching Pennies can be used to investigate the phenomenon of fashion, and thus it is named as the fashion game. The fashion game possesses an obvious mixed Nash equilibrium, yet we are especially interested in pure Nash equilibrium (PNE for short), whose existence cannot be guaranteed. In this paper, we focus on the PNE testing problem, namely given an instance of the fashion game, answer whether it possesses a PNE or not. Our first result is on the negative side: PNE testing, in general, is hard. For the PNE testing problem restricted to several special structures, i.e. lines, rings, complete graphs and trees, either a simple characterization or an efficient algorithm is provided.

Collusion-Resistant Reputation Mechanism for Multi-Agents Systems

Abstract We address the collusion problem in a reputation-driven multi-agent system where agents represent service providers, consumers, and a controller. A game structure is proposed where players are supposed rational and payoff maximizers. The main issue addressed in this paper is how to maintain a collusion-resistant reputation mechanism. We analyze the behavior of different players with respect to the strategies adopted by the opponents. We provide theoretical analysis of the game and discuss the pure and mixed strategy Nash equilibrium along with best response analysis to identify conditions under which the players adopt truthful dominant strategies.

An equilibrium closure result for discontinuous games

For games with discontinuous payoffs Simon and Zame (Econometrica 58:861–872, 1990) introduced payoff indeterminacy, in the form of endogenous sharing rules, which are measurable selections of a certain payoff correspondence. Their main result concerns the existence of a mixed Nash equilibrium and an associated sharing rule. Its proof is based on a discrete approximation scheme “from within” the payoff correspondence. Here, we present a new, related closure result for games with possibly noncompact action spaces, involving a sequence of Nash equilibria. In contrast to Simon and Zame (Econometrica 58:861–872, 1990), this result can be used for more involved forms of approximation, because it contains more information about the endogenous sharing rule. With such added precision, the closure result can be used for the actual computation of endogenous sharing rules in games with discontinuous payoffs by means of successive continuous interpolations in an approximation scheme. This is demonstrated for a Bertrand type duopoly game and for a location game already considered by Simon and Zame. Moreover, the main existence result of Simon and Zame (Econometrica 58:861–872, 1990) follows in two different ways from the closure result.

Some interesting properties of maximin strategies

Since the seminal paper of Nash (1950) game theoretic literature has focused mostly on equilibrium and not on maximin (minimax) strategies. We study the properties of these strategies in non-zero-sum strategic games that possess (completely) mixed Nash equilibria. We find that under certain conditions maximin strategies have several interesting properties, some of which extend beyond 2-person strategic games. In particular, for n-person games we specify necessary and sufficient conditions for maximin strategies to yield the same expected payoffs as Nash equilibrium strategies. We also show how maximin strategies may facilitate payoff comparison across Nash equilibria as well as refine some Nash equilibrium strategies.

A Game-Theoretic Analysis of Hezong and Lianheng-Simultaneous and Sequential Moves

This paper will study a three-player game with infinite periods where each player decides whether to league together to increase his winning rate. We discuss which one will be chosen to be the member of leagues. The results show that Hezong and Lianheng both exist in mixed strategy Nash equilibriums of simultaneous move game. It is possible that the fall of Jin and the Battle of Red Cliffs in Chinese history were just coincident or random events. Furthermore the players usually have a tendency to be in league with the strongest.

Immune Based Approach to Find Mixed Nash Equilibrium in Normal Form Games

Immune Based Approach to Find Mixed Nash Equilibrium in Normal Form Games

Where Strategic and Evolutionary Stability Depart - A Study of Minimal Diversity Games

A minimal diversity game is an n player strategic form game in which each player has pure strategies at his disposal. The payoff to each player is always 1, unless all players select the same pure strategy, in which case all players receive zero payoff. Such a game has a unique isolated completely mixed Nash equilibrium in which each player plays each strategy with equal probability, and a connected component of Nash equilibria consisting of those strategy profiles in which each player receives payoff 1. The Pareto superior component is shown to be asymptotically stable under a wide class of evolutionary dynamics, while the isolated equilibrium is not. On the other hand, the isolated equilibrium is strategically stable, while the strategic stability of the Pareto efficient component depends on the dimension of the component, and hence on the number of players, and the number of pure strategies.

Implementation in Mixed Nash Equilibrium

A mechanism implements a social choice correspondence f in mixed Nash equilibrium if at any preference profile, the set of all pure and mixed Nash equilibrium outcomes coincides with the set of f-optimal alternatives at that preference profile. This definition generalizes Maskin’s definition of Nash implementation in that it does not require each optimal alternative to be the outcome of a pure Nash equilibrium. We show that the condition of weak set-monotonicity, a weakening of Maskin’s monotonicity, is necessary for implementation. We provide sufficient conditions for implementation and show that important social choice correspondences that are not Maskin monotonic can be implemented in mixed Nash equilibrium.

Spatio-temporal credit assignment in population learning

Event Abstract Back to Event Spatio-temporal credit assignment in population learning Johannes Friedrich1*, Robert Urbanczik1 and Walter Senn1 1 University of Bern, Department of Physiology, Switzerland Learning by reinforcement is important in shaping animal behavior, and in particular in behavioral decision making. Such decision making is likely to involve the integration of many synaptic events in space and time. However, using a single reinforcement signal to modulate synaptic plasticity, as suggested in classical reinforcement learning algorithms, a twofold problem arises. Different synapses will have contributed differently to the behavioral decision, and even for one and the same synapse, releases at different times may have had different effects. Here we present a plasticity rule which solves this spatio-temporal credit assignment problem in a population of spiking neurons. The learning rule is spike-time dependent and maximizes the expected reward by following its stochastic gradient. Synaptic plasticity is modulated not only by the reward, but also by a population feedback signal. While this additional signal solves the spatial component of the problem, the temporal one is solved by means of synaptic eligibility traces. In contrast to temporal difference (TD) based approaches to reinforcement learning, our rule is explicit with regard to the assumed biophysical mechanisms. Neurotransmitter concentrations determine plasticity and learning occurs fully online. Further, it works even if the task to be learned is non-Markovian, i.e. when reinforcement is not determined by the current state of the system but may also depend on past events. The performance of the model is assessed by studying three non-Markovian tasks. In the first task, the reward is delayed beyond the last action with non-related stimuli and actions appearing in between. The second task involves an action sequence which is itself extended in time and reward is only delivered at the last action, as it is the case in any type of board-game. The third task is the inspection game that has been studied in neuroeconomics, where an inspector tries to prevent a worker from shirking. Applying our algorithm to this game yields a learning behavior which is consistent with behavioral data from humans and monkeys, revealing themselves properties of a mixed Nash equilibrium. The examples show that our neuronal implementation of reward based learning copes with delayed and stochastic reward delivery, and also with the learning of mixed strategies in two-opponent games. Conference: Computational and Systems Neuroscience 2010, Salt Lake City, UT, United States, 25 Feb - 2 Mar, 2010. Presentation Type: Poster Presentation Topic: Poster session III Citation: Friedrich J, Urbanczik R and Senn W (2010). Spatio-temporal credit assignment in population learning. Front. Neurosci. Conference Abstract: Computational and Systems Neuroscience 2010. doi: 10.3389/conf.fnins.2010.03.00263 Copyright: The abstracts in this collection have not been subject to any Frontiers peer review or checks, and are not endorsed by Frontiers. They are made available through the Frontiers publishing platform as a service to conference organizers and presenters. The copyright in the individual abstracts is owned by the author of each abstract or his/her employer unless otherwise stated. Each abstract, as well as the collection of abstracts, are published under a Creative Commons CC-BY 4.0 (attribution) licence (https://creativecommons.org/licenses/by/4.0/) and may thus be reproduced, translated, adapted and be the subject of derivative works provided the authors and Frontiers are attributed. For Frontiers’ terms and conditions please see https://www.frontiersin.org/legal/terms-and-conditions. Received: 05 Mar 2010; Published Online: 05 Mar 2010. * Correspondence: Johannes Friedrich, University of Bern, Department of Physiology, Bern, Switzerland, friedrich@pyl.unibe.ch Login Required This action requires you to be registered with Frontiers and logged in. To register or login click here. Abstract Info Abstract The Authors in Frontiers Johannes Friedrich Robert Urbanczik Walter Senn Google Johannes Friedrich Robert Urbanczik Walter Senn Google Scholar Johannes Friedrich Robert Urbanczik Walter Senn PubMed Johannes Friedrich Robert Urbanczik Walter Senn Related Article in Frontiers Google Scholar PubMed Abstract Close Back to top Javascript is disabled. Please enable Javascript in your browser settings in order to see all the content on this page.