Bertrand-Edgeworth-Shapley Cycle in a 2 × 2 Game

Taking the detailed balance condition (H. Peythan Young, 2005) or/and the time reversal transformation (Baiesi, Maes and Wynants, 2009) into consideration, we present the Bertrand-Edgeworth-Shapley cycle (Cason, Friedman and Wagener 2005) in a two-dimensional 5 × 5 lattices space based on our data from laboratory experiments. We have 12 independent sessions and in each session 4 Up-Down players and 4 Right-Left players play a 2 × 2 game with the payoff matrix of [(5, 0)(0, 5); (0, 5)(5, 0)] in pairs over 300 periods under random matching protocol. We suggest that, a social Markov jumping in strategy space is a velocity vector, and then it can be measured. By analyzing our experimental data, we find that these vectors form a significant spiral vector field in the 5 × 5 lattices space. Filtering out the detailed balance (or called as traffics process by physicist), we get a clear dynamic pattern (Fig.3 (b, c, d, e)). We also find a ring-mountain-shaped distribution of the time anti-symmetric Markov jumping in the two-dimensional space (Fig.3 (f)). Both algebra matrix analysis and geometrical presentation method reach the same result. These findings, together with the method, might be considerable connections between evolutionary game theory and experimental economics