Decomposing coordination failure in stag hunt games

Abstract

Many situations involve trading-off what is safe for the individual and what is beneficial for the group. This tension is extensively studied as the “Stag Hunt” coordination game. By combining a new theoretical approach with two experiments, this paper shows a disconnection between behavior in Stag Hunt games and the predictions of many models. Any Stag Hunt game can be uniquely decomposed into three payoff components: strategic, behavioral, and kernel. The behavioral component exists in every Stag Hunt game despite being largely ignored by previous models. Arbitrarily many Stag Hunt games exist where these models predict the same behavior. Despite the constant predictions, behavior in repeated and one-shot games systematically vary. The mechanism underlying these results is that all models of Stag Hunt behavior fail to properly account for the influence of the behavioral component.

Appendices

Appendices

Appendix A: Decomposition of general $2 \times 2$ game

Figure 12 shows the decomposition for a general $2 \times 2$ game ${\mathcal {G}}_0$.

Fig. 12

Decomposition of general $2\times 2$ game. The superscripts denote the agent and subscripts (when present) are an index

Without loss, assume (X, X) is payoff dominant and (Y, Y) is the risk dominant. Following Harsanyi and Selten (1988), the notions of payoff and risk dominance are traditionally characterized for game ${\mathcal {G}}_0$ in the following manner:

• (Y, Y) risk dominates (X, X) if $(\pi _{2}^{R}-\pi _{4}^{R})(\pi _{3}^{C}-\pi _{4}^{C}) \ge (\pi _{3}^{R}-\pi _{1}^{R})(\pi _{2}^{C}-\pi _{1}^{C})$.

• (X, X) payoff dominates (Y, Y) if $\pi _{1}^{R} \ge \pi _{4}^{R}$ and $\pi _{1}^{C} \ge \pi _{4}^{C}$ with at least one being a strict inequality.

These concepts can be characterized using the decomposed payoff components from Fig. 12.

• (Y, Y) risk dominates (X, X) if $2s_{2}^{R}\cdot 2s_{2}^{C} \ge 2(-s_{1}^{R})\cdot 2(-s_{1}^{C})$.

• (X, X) payoff dominates (Y, Y) if $b^{R} \ge \frac{(s_{1}^{R} + s_{2}^{R})}{-2}$ and $b^{C} \ge \frac{(s_{1}^{C} + s_{2}^{C})}{-2}$ with at least one being a strict inequality.

Appendix B: Experimental Screenshots

Figure 13 shows the decision screen (top) and feedback screen (bottom) for Experiment 1.

Fig. 13

Screens for Experiment 1

Figure 14 shows the decision screen for Experiment 2. Experiment 2 has the exact same decision screen without the history table at the bottom of the screen and with slight changes to the wording at the top of the screen. There is not feedback between games for Experiment 2.

Fig. 14

Screen for Experiment 2

Appendix C: Additional analysis for Experiment 1

Figure 15 plots the average proportion of subjects who chose the payoff dominant action, X across all 75 periods separated by cohort.

Fig. 15

The average proportion of payoff dominant choices - separated by cohort

Figure 16 reports an analogous figure to Figure 6 except it uses p-values from two-sided Wilcoxon-Mann-Whitney tests testing for differences between the choice of X across Games 1 and 2 in each of the 75 periods. 34 out of 75 periods show significantly different behavior at the 0.10 level. The median p-value across all 75 t-tests (shown in grey) is 0.140.

Fig. 16

P-values from two-sided Wilcoxon-Mann-Whitney tests testing for differences between Games 1 and 2 in each period

Appendix D: Additional intuition for algorithmically creating games

The games for Experiment 2 systematically vary to the values of $s_1$, $|s_2|$, and b. This section provides intuition about how such changes in a game will affect behavior.

Figure 17 shows strategic components $S_1$ and $S_2$ which differ in their $|s_2|$ value.

Fig. 17

$S_1$ and $S_2$ differ in their $|s_2|$ value

$S_2$ has a larger $|s_2|$ value than $S_1$ which means that two facts are true about $S_2$ compared with $S_1$. In $S_2$, both agents earn more in the (Y, Y) outcome ($\bar{S_2} + \alpha > \bar{S_2}$) and both agents earn less when selecting X when the other agent chooses Y ($-\bar{S_2} - \alpha < -\bar{S_2}$). This suggests that Y will be chosen more often when $|s_2|$ is larger.

Figure 18 shows strategic components $S_2$ and $S_5$ which differ in their $s_1$ value.

Fig. 18

$S_2$ and $S_5$ differ in their $s_1$ value

$S_5$ has a larger $s_1$ value than $S_2$ which means that two facts are true about $S_5$ compared with $S_2$. In $S_5$, both agents earn more in the (X, X) outcome ($\bar{S_1} + \alpha > \bar{S_1}$) and both agents earn less when selecting Y when the other agent chooses X ($-\bar{S_1} - \alpha < -\bar{S_1}$). This suggests that X will be chosen more often when $s_1$ is larger.

Figure 19 shows two games which differ in their b value.

Fig. 19

The game on the left and right only differ in their b value

Both games have the same strategic component ($S_1$). The game on the left has the minimum possible behavioral value b1 while the game on the right has a behavioral value that is larger by v. Since the game on the right has a larger b value than the game on the left, two facts are true about the game on the right compared with the game on the left. In the game on the right, both agents earn more in the (X, X) outcome ($\bar{S_1} + b1 + v > \bar{S_1}+ b1$) and both agents earn less in the (Y, Y) outcome ($\bar{S_1} - b1 - v < \bar{S_1}-b1$). This suggests that X will be chosen more often when b is larger.

Appendix E: 40 SH games used in Experiment 2

Appendix F: Additional analysis for Experiment 2

Figure 20 plots the average proportion of payoff dominant choices, X, along with the standard errors in each game. The figure is divided into 10 panels each showing the behavior of a 4-game set; thus each panel shows the behavior in games that only vary in their behavioral value, b. For comparability across sets, b (located on the x-axis) takes on 4 values $b1< b2< b3 < b4$.²⁸

Fig. 20

Average proportion in X for all 40 SH games separated by 4-game set. Standard errors reported

For exposition, allow the set with Game 1 to be called Set 1, the set with Game 2 to be called Set 2, and so on for all 10 sets. Of the 4 games within each set, half of the sets reject the null hypothesis that the groups have the same central tendency (Kruskal-Wallis test p-value $< 0.001$ for Sets 2, 3, 4, 6, and 8). Interestingly, the sets which display different behavior within the set all have high $q^{Nash}$ ($\ge 0.63$) whereas the sets without statistically different behavior have lower $q^{Nash}$ ($\le 0.59$). This relationship is shown in the Table 4.

Table 4 Difference within set