Payoff levels, loss avoidance, and equilibrium selection in the Stag Hunt: an experimental study

1. Payoff levels, loss avoidance, and equilibrium selection in the Stag Hunt: an experimental study Nick Feltovich, University of Houston nfelt@bayou.uh.edu Atsushi Iwasaki, Kyushu University Sobei H. Oda, Kyoto Sangyo University The full paper can be found at www.uh.edu/~nfelt/papers/staghunt.pdf

2. Background When a game has multiple Nash equilibria, game theory does not yield a unique prediction. Example: Stag Hunt R = risky action; S = safe action. Applications: bank runs, adoption of new technology, coordination between firms’ departments, mutinies, football defensemen,… (R,R) and (S,S) are both pure-strategy Nash equilibria. Which—if either—should be expected? (Equilibrium selection)

3. Background Criteria for equilibrium selection typically only look at differences in payoffs between outcomes: • Payoff dominance—differences in equilibrium payoffs • Risk dominance, security, Harsanyi and Selten (1988), Carlsson and van Damme (1993), Haruvy and Stahl (2004)—differences in equilibrium and off-equilibrium payoffs All assume that the absolute level of payoffs does not matter.

4. Background But there is some reason to believe that changes in payoff levels might affect behavior. Theoretical reasons: • Standard expected utility: wealth effects • Expected utility over gains/losses: larger wealth effects • Prospect theory, loss aversion (Kahneman and Tversky (1979)) • Loss avoidance (Cachon and Camerer (1996))

5. Background There is also some empirical evidence that changes in payoff levels can affect behavior—particularly when they cause gains to become losses (or vice versa). • Kahneman and Tversky (1979, etc.): loss aversion in one-shot decision problems (hypothetical, not real, payoffs) • Bereby-Meyer and Erev (1998) and Erev, Bereby-Meyer, and Roth (1999): repeated decision problems • Cachon and Camerer (1996): median- and minimum-effort games • Rydval and Ortmann (2005): one-shot Stag Hunt games • Buchheit and Feltovich (2006): price choices in repeated Bertrand-Edgeworth duopoly If individuals base their decisions (at least partly) on payoff levels, or even if they only believe others do so, there are implications for equilibrium selection.

6. The Stag Hunt games Nash equilibria (all symmetric): Prob(R) = 0, 2/3, or 1. —Unaffected by which version of the game we use.

7. Loss avoidance: predictions Certain-loss avoidance—individuals avoid an action leading to a certain loss if a gain is possible with a different action. —> Prediction: R is more likely to be seen in SHL than in SHH.

8. Loss avoidance: predictions Possible-loss avoidance—individuals avoid an action leading to a possible loss if a different action leads to a certain gain. —> Prediction: R is less likely to be seen in SHM than in SHH.

9. Previous research: Cachon and Camerer (1996) “Must Play” game: mandatory “entry fee” of either 185 or 225.Subjects played 3 rounds of basic game, then 3 of “Must Play (185)” game, then 3 of “Must Play (225)” game.Certain-loss avoidance—>play only 3-7 in “Must Play (185); 5-7 in “Must Play (225)”.Possible-loss avoidance—>play only 2-4 in basic game. Results:—Rounds 1-3: Mean choice=4.59 (45.3% choices of 2-4)—Rounds 4-6: Mean choice=5.35 (100% choices of 3-7, 90.7% choices of 5-7)—Rounds 7-9: Mean choice=6.45 (100% choices of 5-7)

10. Previous research: Rydval and Ortmann (2005) Subjects played each game (and one additional game) once vs. changing opponents.Results:—Game 2 vs. Game 3: More choices of C in Game 3 than Game 2 (consistent with certain-loss avoidance)—Game 4 vs. Game 5: No significant difference in C choices.

11. Experimental design and procedures Subjects played each of the three SH games, in addition to three other games. • Ordering of games is CG-SH-BoS-SH-PD-SH. • Ordering of SH games varies: H-M-L, M-L-H, L-H-M.

12. Experimental design and procedures Treatments: • O: each game played once, complete payoff information. • C: each game played 40 times, random matching, complete payoff information. • R: each game played 40 times, random matching, limited payoff information. • F: each game played 40 times, fixed pairs, limited payoff information. All treatments: random rematching at start of new game.

13. Experimental design and procedures • Actions called R1 or R2 (for row players), C1 or C2 (for column players). • For R and F treatments, correspondence of actions on top/left and bottom/right to risky and safe changed from session to session (R-S or S-R). • Variation of treatment, game ordering, action ordering between subjects. • Variation of game within subjects. • Experiment took place at Kyoto Sangyo University (Kyoto, Japan). • Interaction via Japanese z-Tree • 3 O sessions, 3 C sessions, 6 R sessions, 7 F sessions. • 6-28 subjects per session. • Subjects were paid a fixed showup fee (except in O sessions), plus the results of a randomly-chosen round.

14. Results: one-shot games and first round of C treatment • Risky action more likely in SHL than SHH (but not significant in C treatment) • NOT significantly more likely in SHH than SHM in either treatment • Other Round-1 results: • F and R treatments, all games: Frequency of Top/Left action: 66.5% • Frequency of R (cooperative action) in PD: • —O treatment: 58.3% (42/72) • —C treatment, first round: 68.1% (49/72) • —Combined: 63.2% (91/144)

15. Results: 40-round averages • Significance (Wilcoxon summed-rank test for related samples): • Differences between SHH and SHL significant for F treatment (p<0.01 for pair-level or session-level data) and R treatment (p<0.05 for session-level data). • No significant differences between SHH and SHM (p≈0.11 for session-level data in R treatment).

16. Risky-action frequency: C treatment (all rounds)

17. Risky-action frequency: R treatment (all rounds)

18. Risky-action frequency: F treatment (all rounds)

22. Parametric statistics—methodology • Probit models • Dependent variable: indicator for choice of risky action (1=yes, 0=no) • Independent variables: —round, round2 —SHM indicator, SHM∙round, SHM∙round2 —SHL indicator, SHL∙round, SHL∙round2 —indicators for M-L-H, L-H-M, R-S orderings —previous-round action, previous-round opponent action • Separate estimations for C, R, and F treatments • Individual-subject random effects

23. **: p < 0.05; ***: p < 0.01 Parametric statistics—results (highlights)

24. Determining the effect of the game (SHM or SHL vs. SHH) on risky-action choice • In most (but not all) cases, the variables SHx, SHx∙round, and SHx∙round2 are jointly significant (x = M, L). • But we are mainly interested in their combined effect, which is positively related to the value of the expression βSHx + βSHx∙round∙t + βSHx∙round2∙t2 where t is the round number. • Positive for SHL —> certain-loss avoidance in round t • Negative for SHM —> possible-loss avoidance in round t

25. Estimates of the effect of game on risky-action choice (vs. SHH) based on probit resultsCircles represent point estimates; line segments represent 95% confidence intervals

26. Summary • Both forms of loss avoidance observed, though not in all treatments. • Somewhat stronger evidence of certain-loss avoidance than possible-loss avoidance. • Loss avoidance initially small in magnitude, but becomes more pronounced over time, typically peaking in the second half of the session. Next steps • Effects of information and matching mechanisms. • Results from CG, BoS, PD. • Equilibrium-loss avoidance vs. possible-loss avoidance: are individuals more, less, or equally likely to avoid an action leading to a possible loss if the loss happens in equilibrium?