Hysteresis effects of changing the parameters of noncooperative games

We adapt the method used by Jaynes to derive the equilibria of statistical physics to instead derive equilibria of bounded rational game theory. We analyze the dependence of these equilibria on the parameters of the underlying game, focusing on hysteresis effects. In particular, we show that by gradually imposing individual-specific tax rates on the players of the game, and then gradually removing those taxes, the players move from a poor equilibrium to one that is better for all of them.

Figure 1

Col's mixed strategy plotted against $\vec{\beta }$. This is for the QRE of the game in Eq. (9). The hysteresis path discussed in the text is highlighted, starting at the bottom right.

Figure 2

$\mathbb{E}\left(u_{\text{col}}\right)$ vs $\vec{\beta }$ under the QRE of the game in Eq. (9). The hysteresis path discussed in the text is highlighted, starting at the bottom right.

Figure 3

Col's expected post-tax utility plotted against the tax rates (expressed as fractions) on the two players. This is for the QRE of the game in Eq. (9). The constraint for each player $i$ in this taxed game is $b_i=5$. So for example, $\vec{\alpha }=(1,1)$ corresponds to the point with $\vec{\beta }=(5,5)$ in Fig. 2.

Figure 4

A QRE surface with paths shown for the anarchy (red), socialism (blue), and market (purple) procedures. As in Fig. 2, the $x$ and $y$ axes are player rationalities, ${\beta }_{\text{row}}$ and ${\beta }_{\text{col}}$, and the $z$ axis is expected utility (this time of player row). This plot is for a variant of the battle of the sexes game where all utility functions are multiplied by $-1$.

Figure 5

The difference between the discounted sums of future expected utilities of the two players under the “socialism” and “market” procedures, plotted against the discounting factor $\gamma$.

Figure 6

The expected utility of the row player along the path through $\vec{\beta }$ highlighted in Fig. 2, illustrated as a function of the row player's rationality ${\beta }_{\text{row}}$. The path starts at the bottom right, then travels left, before turning and finishing at the top right.