Indeed, for cell B,A 40 is the maximum of the first column and 25 is the maximum of the second row. For A,B 25 is the maximum of the second column and 40 is the maximum of the first row. Same for cell C,C. For other cells, either one or both of the duplet members are not the maximum of the corresponding rows and columns.
Darwinian Selection and Non-existence of Nash Equilibria. We study selection acting on phenotype in a collection of agents playing local games lacking Nash equilibria. After each cycle one of the agents losing most games is replaced by a new agent with new random strategy and game partner.
The network generated can be considered critical in the sense that the lifetimes of the agents is power law distributed. The longest surviving agents are those with the lowest absolute score per time step. The emergent ecology is characterized by a broad range of behaviors. Nevertheless, the agents tend to be similar to their opponents in terms of performance.
Mutations act at random at the level of genotype while selection acts at the phenotype. Darwinian selection is expected to occur because some phenotypes are more viable than others .
The viability or fitness of a given Nash equilibrium existence is, of course, not an absolute quantity but depends on context and the environment the phenotype is exposed to .
The environment physical as well as biotic is to a large extent produced by co-existing phenotypes, and hence the environment and the corresponding fitness are self-organized emergent properties of the ecology.
Moreover, fitness or vigor is necessarily a relative quantity. An organism can become so vigorous that it removes its own foundation of existence; a balance very relevant e.
For organisms to be able to coexist for extended periods of time they need to develop phenotypes which are in a kind of restrained poise.
Here we analyze Darwinian selection acting on phenotype in a model consisting of an adaptive network of agents of different strategies mutually competing in local zero-sum games lacking Nash equilibria. We let selection act by removing the agents with the smallest score most negative after a round of games.
Thus the viability of an agent is a function of his phenotypical behavior in a specific environment. When the local games are of opposing interest with only a single winner we find that the agents organize into a system where agents are homogeneous in terms of success but heterogeneous in terms of activity.
The agents achieve longevity by organizing themselves in a way that minimize the absolute value of their score. By this strategy they avoid to be amongst the worst performers themselves; moreover low absolute score allows the agents to retain well tuned partners by not forcing many lost games on to their opponents, which would lead to the elimination of the partner.
Our model is closely related to previous network models. Kauffman introduced Boolean networks to study the relationship between evolution and self-organization .
Aspects of Darwinian evolution were added to the minority game by Zhang . This line of approach were further developed by Paczuski, Bassler and Corral PBC who considered agents playing the minority game . The goal of agents is to be among the global minority though they are linked only to a subset of all the agents.
The strategy of the worst performing agent is randomly changed after each cycle while the links of the network remain unchanged .
Here we modify the model introduced by PBC to a Local Darwinian Network model LDN in which we allow agents to play competitive local games and to adapt their connections as well as their strategies.
Viewed from the perspective of evolutionary ecology it appears to be more natural to consider agents performing local games organisms or species entangled in a web of mutual competitions. In either case it is natural to update the set of interaction links. Further, from a statistical mechanics or complex systems perspective we expect systems coupled globally to more readily enter a critical state, hence it is of interest to explore the criticality of purely local systems.
It follows that any discrepancy in the switching behavior of two competing agents is of dynamical origin. The dynamics consists of two types of updates: Each agent is defined as aggressor A of one other randomly assigned agent thus agents participate on average in two games who acts as opponent O in a zero sum game.
The game has the property that, when considering only pure non-randomized strategies, there exists no Nash equilibrium. No Nash equilibrium exists in the sense that it will always be possible for either player to benefit by changing his strategy if the opponent sticks to his strategy.
In our simulations, two agents are never allowed to act as mutual aggressors in which case both agents could receive a zero score. Neither do we allow agents to act as aggressor for themselves.
The fact that agents cannot collaborate in order to achieve neutral zero scores will be referred to as an opposing interests property of the network.
The signals used for the zero-sum games are generated in the following way. One can think of a specific choice for the functional form of fi as the genotype of agent number i.
The K source-agents associated with i can include the opponent of i but will not include agent i. The source-agents and the opponents are to be thought of as the environment of an agent. For fixed assignments of source-agents as well as aggressor and opponents Eq. Since the state space of the net is finite and the dynamics is deterministic, periodic orbits will always be reached though the maximum length of a period is huge 2N.
We measure the average score per time step of the agents over a round of games consisting of either simulating the transient plus one periodic cycle or by performing parallel updates which ever is the shortest.Theorem 1 (Nash, ) There exists a mixed Nash equilibrium.
Here is a short self-contained proof. We will deﬁne a function Φ over the space of mixed strategy proﬁles. 1 Existence of Nash Equilibrium The first existence theorem is due to Nash and dates back to Many generalizations appeared since then.
Finding less and less restrictive sufficient conditions have been an active field of research [Forgo et al., ]. The proofs of. Reny™s () equilibrium existence theorem states that every compact, quasiconcave, better-reply secure game has a Nash equilibriumin pure strate-gies.
Theorem 1 (Reny,) If G = (X i;u i) i2I is compact, quasiconcave, and better-reply secure, then it possesses a pure strategy Nash equilibrium. Keywords: Noncooperative game, strong Nash equilibrium, coalition, weak Pareto-efﬁciency. 1 Introduction This paper studies the existence of strong Nash equilibrium (SNE) in general economic games.
Although Nash equilibrium is probably the most important central behavioral solution concept in game theory, it has some drawbacks.
@John Baez existence of a Nash equilibrium for two person finite zero sum games is a linear programming problem. The existence of symmetric equilibrium for a two person finite game with symmetric payoff matrices that are symmetric is a quadratic programming problem.
For example, John Nash's original proof of the existence of a Nash equilibrium, from , was such an existence theorem. In a constructive approach was found. In a .