^{1, 2}

^{2}

^{1}

^{2}

The paper is a theoretical investigation into the potential application of game theoretic concepts to neural networks (natural and artificial). The paper relies on basic models but the findings are more general in nature and therefore should apply to more complex environments. A major outcome of the paper is a learning algorithm based on game theory for a paired neuron system.

Individual neurons are the building blocks for more complex neural circuits. In natural systems these more complex neural circuits interact with other components in a manifold of ways thereby generating the compellingly sensual world of behavior around us. Although tireless and tedious efforts in various disciplines culminated in fundamental insights in the field, there are still many unknowns about individual neurons and the processes in which individual neurons interact and organize themselves in neural circuits (e.g., [

Recently, game theory has obtained some attention in the field of neuroscience. The field of neuroeconomics, for instance, combines the two fields in experiments with human and nonhuman players in order to better understand human decision-making (e.g., [

In the remainder of this text, Section

Our previous work in various areas (e.g., artificial intelligence, soft computing, reasoning under uncertainty, and neuroscience) identified that many cooperations between two agents (artificial or natural) can be interpreted or bear some of the characteristic features of a game. For example, the main concepts in a game are the players in a game, a set of rules by which the game is played, and an outcome in the form of a reward or a punishment (more generally referred to as a payoff) for the players in the game. In addition, a so-called payoff matrix is a common scheme to represent the dynamic behavior of a game.

Figure

Relationships between (a) biological neurons, (b) game theory, and (c) artificial neurons.

Laying this issue aside, it is possible to provide a rather straightforward mathematical description for the modeling of the global behavior desired for the two neurons in Figure

Collectively, it is possible to think of Neuron-1 and Neuron-2 as a simple input-output unit that behaves similar to a switch. In terms of its global behavior, a perceptron can be interpreted exactly in the same way. (It is not necessary to elaborate on the perceptron learning algorithm in great detail as this information is widely available in the neural network literature (e.g., in [

In order to appreciate the forthcoming sections and to avoid unnecessary confusion, it is helpful to understand that game theory distinguishes between different types of games. At large, there are

It is tempting now to immediately view and deal with Figure

Many of the formalisms in traditional game theory imply a degree of rationality by the players/agents involved in a game. As crucial as the notion of rationality is for the theory, the term rationality is not without problems. For one thing, the term rationality is not universally defined, and for another thing, human agents are often not the hyper-rational agents the theory requires them to be. Many applications of game theory therefore involve abstractions and simplifications to various degrees. For instance, this happens when game theory is applied to the modeling of interactions in genes, viruses, or cells, as is the case in

These terms are problematic too and can quickly lead into a deep philosophical discussion. A root problem in Figure

Imagine that for some reason Neuron-1 and Neuron-2 in Figure

A static game with complete information and mixed strategies.

The remaining text in Section

For simplicity, Figure

Viewpoint of Player-1 (Neuron-1).

According to Figure

Similarly, the expected payoff for Player-1 for playing the pure strategy Rest is

Figure

Decision-making support for Player-1 if Player-1 believes that Player-2 plays the mixed strategy

It is also possible to consider mixed strategy responses by Player-1. Player-1's expected payoff

What exactly is at stake here? At stake is the goal to maximize the payoff for Player-1 expressed by (

Player-1's best response (maximizing the expected payoff

All in all, Figure

This section describes Player-2's view from Figure

The expected payoff for Player-2 for playing the pure strategy Rest is

Figure

Decision-making support for Player-2 if Player-2 believes that Player-1 plays the mixed strategy

Further, Player-2's expected payoff

The interpretation of (

Player-2's best response (maximizing the expected payoff

Figure

Figures

Combined view of best responses for Player-1 and Player-2. The three intersections between

The interesting features in Figure

In the communicating neuron context of Figure

At this moment, it may be useful to take a step back and to evaluate the results mentioned before a bit more carefully. The results are derived from a purely formal investigation of the (arbitrary) game illustrated in Figure

This section concentrates on dynamic games with complete and perfect information. Such games have three distinctive features: (i) the moves in the game occur sequentially, (ii) a sort of move history exists (i.e., all previous moves are observed before a next move is chosen), and (iii) the payoffs in the payoff matrix are known to all players in the game. Remember, a game has complete information if the content of the payoff matrix is common knowledge to all players in the game. A game has

A game tree for a simple two-move game. There are two players (1 and 2) and the numbers at the bottom of the tree represent the payoff for each player traversing a particular path.

The strategies for the players in Figure

Player one decides on one of the available strategies (here,

Player two observes this decision and decides on an appropriate strategy response (here,

The players receive their payoffs.

Backwards induction works its way up from the bottom of the tree. Assume the position at the bottom of Path-1 where player one has decided to play strategy

Figure

Game tree for the communicating neuron example (Figures

For the game tree in Figure

In order to acquire a capacity for decision-making, a network has to evolve from an unorganized state to an organized (synchronized) state with the latter state demonstrating the desired problem-solving potential. The mechanism that drives artificial neural networks from an unorganized state to an organized state is typically realized by a learning algorithm. This section describes a learning algorithm based on game theory for artificial neural networks. The question marks in Figure

A learning algorithm for an artificial neural network based on game theory may exploit the payoffs in the matrix (i.e., the payoff function) and the mixed strategies.

For the algorithm, imagine a one-dimensional, linearly separable, and supervised learning classification task. Figure

A one-dimensional, linearly separable, and supervised learning classification task.

The game theory: based learning algorithm at work. Neuron-1's point of view: (a), (b), and (c). Neuron-2's point of view (d).

The figures in Figure

Figure

This section initially repeats an important fact that has been mentioned several times in this text already, namely, that the scopes for neuroscience and game theory are quite rich and rather complex in their own right, and that this paper, consequently, can only present a condensed view of the many challenges involved in the wider context of this investigation. A second important statement in this section is the finding that although there is work combining game theory and neuroscience, according to our understanding, the two fields have not been combined in the way presented in this paper. For example, the relatively young field of neuroeconomics combines the two fields in experiments with human and nonhuman players (e.g., see Sanfey et al. [

The possible application of game theory to fields such as human computer interaction indicates that game theory has long left its traditional environment–-economy and human decision-making (the famous mathematician John Forbes Nash was awarded, jointly, the Nobel Prize in Economics in 1994 for his work in game theory). Today, the theory is widely applied in the natural sciences for the modeling of a rich variety of biological games involving agents of various types. Indeed, the principles of the theory are general enough to attract cutting-edge research in artificial intelligence or systems biology in applications where web-based intelligent agents or robots may have to wrestle with complex decision-making problems [

Although it is clear that several other interesting studies could be mentioned here, this review section wants to draw to an end by commenting, briefly, on the timing of games. This paper dealt with static and dynamic games in a separate way and this treatment may have given the impression that a system, over time, always sticks to one type of game, which is questionable. Consider the timing of games in a different context. Take a tournament where the teams

The paper presented a novel concept for describing individual neurons under the game theoretic framework. The paper created a firm understanding about some of the fundamental problems in game theory and emphasized that these problems are not unique to the domain of neural systems, but that these problems reach out more deeply into game theory, science, and the world around us. The paper demonstrates that various strategic game theoretic concepts and calculations seem to be naturally suitable for the modeling of the behavior of a paired neuron system (and possibly for more complex networks). This finding was further solidified through the specification of a novel learning algorithm based on game theory for the purpose of neural learning.

The first author gratefully acknowledges the support of Japan Society for the Promotion of Science (JSPS Fellowship no. S09168).