The Undecidability of the Spatialized Prisoner's Dilemma

Theory and Decision, 1997

Often a question arises as to what the eventual outcome of an initial spatial configuration of strategies will be: Will a single strategy prove triumphant in the sense of progressively conquering more and more territory without opposition, or will an equilibrium of some small number of strategies emerge? Here it is shown, for finite configurations of Prisoner's Dilemma strategies embedded in a given infinite background, that such questions are formally undecidable: there is no algorithm or effective procedure which, given a specification of a finite configuration, will in all cases tell us whether that configuration will or will not result in progressive conquest by a single strategy when embedded in the given field. The proof introduces undecidability into decision theory in three steps: by (1) outlining a class of abstract machines with familiar undecidability results, by (2) modelling these machines within a particular family of cellular automata, carrying over undecidability results for these, and finally by (3) showing that spatial configurations of Prisoner's Dilemma strategies will take the form of such cellular automata.

Abstract. The prisoner's dilemma is a two-player non-zero-sum game. Its iterated version has been frequently used to examine game strategy evolution in the literature. In this paper, we discuss the setting of neighborhood structures in its spatial iterated version. The main characteristic feature of our spatial iterated prisoner's dilemma game model is that each cell has a different scheme to represent game strategies.

2007

The Prisoner's Dilemma (PD) is one of the most popular games of the Game Theory due to the emergence of cooperation among competitive rational players. In this paper, we present the PD played in cells of one-dimension cellular automata, where the number of possible neighbors that each cell interacts, z, can vary. This makes possible to retrieve results obtained previously in regular lattices. Exhaustive exploration of the parameters space is presented. We show that the final state of the system is governed mainly by the number of neighbors z and there is a drastic difference if it is even or odd.

Springer Proceedings in Mathematics & Statistics, 2021

The prisoners dilemma (PD) is a game-theoretic model studied in a wide array of fields to understand the emergence of cooperation between rational self-interested agents. In this work, we formulate a spatial iterated PD as a discrete-event dynamical system where agents play the game in each time-step and analyse it algebraically using Krohn-Rhodes algebraic automata theory using a computational implementation of the holonomy decomposition of transformation semigroups. In each iteration all players adopt the most profitable strategy in their immediate neighbourhood. Perturbations resetting the strategy of a given player provide additional generating events for the dynamics. Our initial study shows that the algebraic structure, including how natural subsystems comprising permutation groups acting on the spatial distributions of strategies, arise in certain parameter regimes for the payoff matrix, and are absent for other parameter regimes. Differences in the number of group levels in the holonomy decomposition (an upper bound for Krohn-Rhodes complexity) are revealed as more pools of reversibility appear when the temptation to defect is at an intermediate level. Algebraic structure uncovered by this analysis can be interpreted to shed light on the dynamics of the spatial iterated PD.

Journal of Combinatorial Theory, Series A, 2013

We study two-player take-away games whose outcomes emulate two-state one-dimensional cellular automata, such as Wolfram's rules 60 and 110. Given an initial string consisting of a central data pattern and periodic left and right patterns, the rule 110 cellular automaton was recently proved Turing-complete by Matthew Cook. Hence, many questions regarding its behavior are algorithmically undecidable. We show that similar questions are undecidable for our rule 110 game.

EPL (Europhysics Letters), 2009

We study co-evolutionary Prisoner's Dilemma games where each player can imitate both the strategy and imitation rule from a randomly chosen neighbor with a probability dependent on the payoff difference when the player's income is collected from games with the neighbors. The players, located on the sites of a two-dimensional lattice, follow unconditional cooperation or defection and use individual strategy adoption rule described by a parameter. If the system is started from a random initial state then the present co-evolutionary rule drives the system towards a state where only one evolutionary rule remains alive even in the coexistence of cooperative and defective behaviors. The final rule is related to the optimum providing the highest level of cooperation and affected by the topology of the connectivity structure.

Brazilian Journal of Physics, 2008

The Prisoner's Dilemma (PD) is one of the most popular games of the Game Theory due to the emergence of cooperation among competitive rational players. In this paper, we present the PD played in cells of onedimension cellular automata, where the number of possible neighbors that each cell interacts, z, can vary. This makes possible to retrieve results obtained previously in regular lattices. Exhaustive exploration of the parameters space is presented. We show that the final state of the system is governed mainly by the number of neighbors z and there is a drastic difference if it is even or odd.

IEEE Transactions on Evolutionary Computation, 2001

The prisoner's dilemma (PD) involves contests between two players and may naturally be played on a spatial grid using voter model rules. In the model of spatial PD discussed here, the sites of a two-dimensional lattice are occupied by strategies. At each time step, a site is chosen to play a PD game with one of its neighbors. The strategy of the chosen site then invades its neighbor with a probability that is proportional to the payoff from the game. Using results from the analysis of voter models, it is shown that with simple linear strategies, this scenario results in the long-term survival of only one strategy. If three nonlinear strategies have a cyclic dominance relation between one another, then it is possible for relatively cooperative strategies to persist indefinitely. With the voter model dynamics, however, the average level of cooperation decreases with time if mutation of the strategies is included. Spatial effects are not in themselves sufficient to lead to the maintenance of cooperation.

Journal of Theoretical Biology, 1995

The iterated Prisoner';s Dilemma is the standard model for the evolution of cooperative behavior in a community of egoistic agents. Within that model, a strategy of 'tit-for-tat' has established a reputation for being particularly robust. Nowak and Sigmund have shown ...

Journal of Artificial Societies and Social Simulation

In the iterated prisoner's dilemma game, new successful strategies are regularly proposed especially outperforming the well-known tit_for_tat strategy. New forms of reasoning have also recently been introduced to analyse the game. They lead William Press and Freeman Dyson to a double infinite family of strategies that-theoretically-should all be e icient strategies. In this paper, we study and confront using several experimentations the main strategies introduced since the discovery of tit_for_tat. We make them play against each other in varied and neutral environments. We use the complete classes method that leads us to the formulation of four new simple strategies with surprising results. We present massive experiments using simulators specially developed that allow us to confront up to , strategies simultaneously, which had never been done before. Our results show without any doubt the most robust strategies among those so far identified. This work defines new systematic, reproductible and objective experiments suggesting several ways to design strategies that go a step further, and a step in the so ware design technology to highlight e icient strategies in iterated prisoner's dilemma and multiagent systems in general.