Preference and Belief Imprecision in Games

1. Preference and Belief Imprecision in Games David Butler Andrea Isoni Graham Loomes Daniel Navarro-Martinez Workshop on Noise and Imprecision in Individual and Interactive Decision-Making Warwick, 16–18 April 2012

2. Introduction and motivation (1) Why do the predictions of game theory fail systematically when a game is faced for the first time? (Kagel and Roth 1995; Goeree and Holt 2001; Camerer 2003) Standard assumptions: • Payoffs represent utilities • Utilities are known precisely • Other players’ payoffs are used to form beliefs about what they will do • Players are self-interested, risk-neutral expected utility maximisers Standard predictions: • Equilibrium actions and beliefs • Consistency between beliefs and actions

3. Introduction and motivation (2) QRE • Relaxes the assumption that utilities are known precisely • Retains the general structure and emphasis on equilibrium Level-k and CH models • Relax the assumption that players have equilibrium beliefs • Retain the maximisation (best response) assumption Our study • Tries to measure the extent to which beliefs and preferences over actions are imprecise • Explores whether failure to best-respond is systematically related to preferences and/or belief imprecision

4. Experimental design (1) Basic ingredients • Elicit beliefs and belief imprecision using simple tasks • Obtain some measure of preference imprecision • Focus on 2x2 games • Study classes of games in which level-k theory and QRE make distinctive comparative-static predictions • Controls for other factors (e.g. strategic uncertainty, social preferences)

5. Experimental design (2) Representation of games (“interactions”)

6. Experimental design (3) Belief elicitation task • “Imagine that the other person is drawn from a group of 20 people” • Lower estimate: “smallest number of these 20 people who are likely to choose Left” • Upper estimate: “largestnumber of these 20 people you can reasonably imagine choosing Left” • Best estimate (≥ lower estimate and ≤ upper estimate)

7. Experimental design (4) Reward mechanism “We ask you to be as accurate as you can in reporting your estimates. In order to reward you for doing this, we will do the following. After the experiment is finished, we will select one of the 12 interactions at random, and we will pay £10 to each person whose best estimate exactly matches the number of participants who actually chose Left out of the first 20 who participated in the experiment. This £10 is in addition to anything you receive at the end of the session today. Those who get the extra £10 will be notified by email. To insure the transparency of the whole process while preserving the privacy of these winners, we will email a list of the university registration numbers of the winners to all participants.”

8. Experimental design (5) Choice and confidence task • Choose top: one of the six buttons in the left box • Choose bottom: one of the six buttons in the right box • Indifference not allowed • Increasing confidence further away from the middle Reward mechanism “the computer will randomly select one of the 12 interactions for you and the person you have been paired with. This interaction will be shown on your screen again. The choices that you and the other person made in that interaction will determine how much each of you will get paid.”

9. Experimental design (6) Treatments Joint Separate Beliefs and choices on the same Choices for all games, then beliefs screen for each game for all games Part 1 (beliefs & choices)Part 1 (choices) Part 2 (beliefs) • Choice/confidence bar appears after beliefs are entered • Beliefs can be changed while entering choice

10. The games (1) Row Col • QRE • Gradual change in probabilities moving from MP to AMP3 • Prob(T) decreases • Prob(L) increases Matching pennies games MP AMP1 AMP2 AMP3

11. The games (2) Row Col Stag hunt games SH1 SH2 • QRE • Probability of T (L) always below 0.5 in SH1 and always above 0.5 in SH2 • risk aversion could complicate the picture – reducing both probabilities, possibly even bringing down below 0.5 in SH2

12. Results: Beliefs (1) Matching pennies game

13. Results: Beliefs (2) Asymmetric MP games

14. Results: Beliefs (3) Stag hunt games

15. Results: Beliefs (4) To sum up: • Subjects seem to understand belief elicitation task (see MP game) • Lower and upper estimates respond sensibly to the parameters of the games (compare player roles in AMP games) • Belief ranges vary somewhat across games (narrower ranges for SH) • Belief ranges indicate that, even in simple 2x2 games, elicited beliefs may be substantially imprecise • In some classes of games (MP vs. AMP) we observe discontinuities in belief distributions that are consistent with level-k thinking, but not in others (SH games)

16. Results: Choices & confidence (1) Matching pennies games • Low confidence when decisions are highly unpredictable (higher for salient cell?) • Choices react to changes in own payoffs (inconsistent with level-k if levels are stable across games) – confidence higher for more popular choices • Choices do no react much to changes in payoffs of opponent – confidence higher for less popular choices

17. Results: Choices & confidence (4) Stag hunt games • Very similar behaviour in both games for both roles (contrary to level-k theory and QRE) • Higher confidence oncooperativechoice, especially for row players

18. Results: Choices & confidence (5) To sum up: • Subjects make substantial use of the confidence instrument • Higher confidence is found for choices that are (i) very popular and/or (ii) distinctively good for both players and/or (iii) salient • The discontinuities predicted by level-k theory for (A)MP games are less sharp than in belief data, and absent for SH games, suggesting that the two tasks may not be obviously related in subjects’ minds

19. Results: Best responses (1) Stag hunt games Matching pennies games Rank correlation between (strong) best response rates and confidence = 0.188* No strong link between belief ranges and best response rates

20. Concluding remarks (1) Our measures of beliefs and confidence vary systematically and show sensible patterns within and between the different classes of 2x2 games that we have considered The extent to which strategy choices represent best responses to stated beliefs varies somewhat across games, but: • Best responses are slightly more common when beliefs and strategy choices are elicited simultaneously than separately • Belief imprecision, as measured by the average difference between UE and LE, does not seem to be strongly related to rates of best responses, but ranges are relatively narrow in the games with the highest best response rates • Higher confidence levels are associated with higher best response rates

21. Concluding remarks (2) Beliefs and behaviour are much more diverse than deterministic level-k model would entail: • Different tasks (belief vs. choice) prompt level-k thinking to a different extent • Keeping the ‘frame’ constant, some classes of games (AMP) seem more conductive to level-k thinking than others (SH) (whether there are ‘co-operative’ outcomes may be a factor) Our results show that eliciting measures of confidence and belief imprecision can add interesting dimensions to the experimental analysis of strategic interactions

22. The end Thank you!

23. Experimental design (7) Other tasks In addition to the belief and choice/confidence tasks, there were two more types of tasks presented in two (counterbalanced) blocks, which always came at the end: • Multiple choices between Top and Bottom with both players’ payoffs,in which the pre-set probability of Left increased (decreased) in 10% steps from 0% (100%) to 100% (0%) • Multiple choices between Top and Bottom with own payoffs only, in which the pre-set probability of Left increased (decreased) in 10% steps from 0% (100%) to 100% (0%) Probabilities implemented by drawing 1 of 100 numbered discs Results not reported here

24. The games (2) Row Col ‘Battle of the sexes’ games BS1 BS2 BS3 • QRE • Small change in probabilities between BS2 and BS3 Note: BS3 is like a SH game

25. The games (3) Row Col Prisoner’s dilemma games PD1 PD2 PD3 • QRE • Gradual increase in probability of T (L) moving from PD1 to PD2 to PD3

26. Results: Beliefs (3) Battle of the sexes games

27. Results: Beliefs (4) Prisoner’s dilemma games

28. Results: Choices & confidence (2) Battle of the sexes games • Strategy conducting to preferred NE played more frequently (slightly higher confidence on salient cell) • T-L played increasingly less frequently and no discontinuity between BS2 and BS3 (contrary to level-k) • Very high confidence on Pareto-superior equilibrium (remember BS3 is a SH game)

29. Results: Choices & confidence (3) Prisoner’s dilemma games • Unexpected difference in behaviour between two player roles (they saw exactly the same table, beliefs were no different) • T-L played increasingly more frequently (contrary to level-k, in which dominance strategy should be played in all cases) • Very high confidence on both the NE and the co-operative solution when the two are not very different

30. Results: Best responses (1) Prisoner’s dilemma games Stag hunt games Matching pennies games Battle of the sexes games Rank correlation between (strong) best response rates and confidence = 0.188* No strong link between belief ranges and best response rates

31. Results: Best responses (2) AMP games – Level-k and best responses

32. Results: Best responses (3) BS games – Level-k and best responses

33. Results: Best responses (2) PD games – Level-k and best responses

34. Results: Best responses (2) SH games – Level-k and best responses