Classification by decomposition: a novel approach to classification of symmetric $$2\times 2$$ games

Abstract

In this paper, we provide a detailed review of previous classifications of $$2\times 2$$ 2 × 2 games and suggest a mathematically simple way to classify the symmetric $$2\times 2$$ 2 × 2 games based on a decomposition of the payoff matrix into a cooperative and a zero-sum part. We argue that differences in the interaction between the parts is what makes games interesting in different ways. Our claim is supported by evolutionary computer experiments and findings in previous literature. In addition, we provide a method for using a stereographic projection to create a compact 2-d representation of the game space.

Teamwork Management in Additive Manufacturing Using Game Theory

“…The strategies are the actions available to the players [23]. Böörs et al [25] have reported that any game can be broken down into a mutual interest game and a zero-sum game, according to Kalai et al [26]. A two-player game with payoff bimatrix P can always be decomposed into the sum of a zero-sum game with payoff bimatrix Z and a game of mutual interest with payoff bimatrix C : P = C + Z. Two-player game bimatrix may be decomposed into one mutual interest game and a zero-sum game.…”

“…The absolute value of the variables C 1 , C 2 and Z 1 indicate the strengths of the players' preferences. The conflict part counteracts the mutual interest part of the decomposition, i.e., the conflict variable Z 1 and at least one of the mutual interest variables C 1 or C 2 present opposite signs, and the value of Z 1 is greater than this variable [25].…”

“…These can be characterised as close and distant neighbours, see Figure 7(a). This depends on how many swaps are required to turn one into the other, see Figure 7(b) [25,28].…”