Virtual bargaining as a micro-foundation for social behaviour

1. Virtual bargaining as a micro-foundation for social behaviour Nick Chater(with TigranMelkonyan Jennifer Misyak& HossamZeitoun) Behavioural Science Group

2. TOPIC OVERVIEW • Nash equilibria: Too many or too few? • Weakly feasible equilibrium: Expanding the set of equilibria • Choosing an equilibrium by virtual bargaining: Reducing the set of equilibria • Communication as virtual bargaining • Applications and future directions

3. 1. NASH EQUILIBRIA: TOO MANY OR TOO FEW?

4. Nash equilibrium • A pair of strategies such that each player’s strategy is the best response to the other • The problem of multiple equilibria • E.g., Hi-Lo game 2 pure strategies: Hi, Hi (10, 10) Lo, Lo (1, 1) 1 mixed strategy: (10/11 Lo, 1/11 Hi), (10/11 Lo, 1/11 Hi) (,)

5. Nash equilibrium • Battle of the sexes (asymmetric version) 2 pure strategies: Ballet, Ballet: (1, 9) Football, Football: (10, 8) 1 mixed strategy: (8/17 B, 9/17 F), (10/11 B, 1/11 F): (, 4)

6. Multiple Nash equilibria are problematic • How can we know which equilibrium to ‘best respond’ to? • If I think Hi, Hi; you think Lo, Lo, we’re in trouble!

7. A long tradition of ‘refinements’ of Nash (to prune ‘bad’ equilibria) • Trying to prune the number of equilibria • Pay-off dominance • Risk dominance • … • And for dynamic games • Sub-game perfection • …

8. But is Nash equilibrium also too restrictive? • The boobytrap game (Misyak & Chater, in press) • Traveller’s Dilemma (Basu, 1994)

9. The Boobytrap Game * Do nothing? * Buy boobytrapfor own safe? * Steal? (will inevitably lose/damage some of the ‘treasure’)

10. The Boobytrap Game Prisoner’s Dilemma, with its demoralizing D,D Nash equilibrium

11. The Boobytrap Game Prisoner’s Dilemma – but now with an extra move

12. The Boobytrap Game According to standard game theory, B is dominated by C; so Nash equilibrium is still DD

13. The Boobytrap Game But if people buy the boobytrap (or even better, buy it sometimes) shouldn’t 29, 29 be attainable?

14. The Traveller’s Dilemma (Basu, 1994) • Each player chooses a sum of money between $1 and $100 • Both get the payoff, in $, associated with the lowest number • If one player’s number is strictly lower, then transfer $2 from the ‘greedy’ to the ‘modest’ player

15. The Traveller’s Dilemma (Basu, 1994) • E.g., $100, $100  $100, $100 $100, $99  $99-$2=$97, $99+$2=$101 $1, $100  $1+$2=$3, $1-$2=-$1 • Unique Nash equilibrium (!) $1, $1  $1, $1 But this seems like a terribly bad outcome! Shouldn’t something near $100, $100 be attainable?

16. 2. WEAKLY FEASIBLE EQUILIBRIUM: EXPANDING THE SET OF EQUILIBRIA

17. Key shift: from ‘I’ to ‘we’ • Don’t ask: what shall Ido in response to your move? (I don’t know your move, anyway) • Do ask: what could we agree to do? • No external mechanism for enforcing the agreement; and no assumption of ‘trust.’

18. Who do we trust? • Nash assumes (implicitly) that we don’t trust ourselves(we may violate our side of the bargain, and best respond) vs • In making a bargain, we trust ourselves, but we don’t (necessarily) trust the other player (the other may violate their side of the bargain and best respond)

19. Each player is cautious • Neither knows whether the other will • “Go through” with the ‘bargain’ • Best-respond (if different) • Call the ‘sure thing’ payoff the minimum of these • Suppose each player aims to maximize the ‘sure thing’

20. Weakly feasible equilibrium (WFE) • A pair of strategies, is a WFE if: • Player 1 can’t obtain a better “sure thing,” by shifting to some other strategy, given that player 2 plays • Player 2 can’t obtain a better “sure thing,” by shifting to some other strategy, given that player 1 plays

21. In symbols • Sure thing for player i , for strategies : where is the best response of player –i to • A pair of strategies, is a WFE if, for all ,

22. WFE extends Nash Nash equilibrium WFE But the extra equilibria are one’s that seem very natural to play…

23. Application to the Boobytrap Game B, B is (nearly) a WFE: if I buy the boobytrap, you can either best respond (C) or go through with the bargain (B). Either way, I get 29.

24. Actually, I only need to buy the boobytrapenough to deter D… …so a mixed strategy of C and B is best But notice this is a dominated by C (and hence not a Nash equilibrium)

25. The Traveller’s Dilemma (Basu, 1994) • Each player chooses a sum of money between $1 and $100 • Both get the payoff, in $, associated with the lowest number • If one player’s number is strictly lower, then transfer $2 from the ‘greedy’ to the ‘modest’ player • Then, many WFE: ($1, $1), ($2, $2)… ($100, $100)

26. The Traveller’s Dilemma (Basu, 1994) ($100, $100) Has a sure thing for Player 1: Min($98, $100) =$98 Can the first player get a better sure thing? No! For any other n<100, the pair of strategies: ($n, $100) Has a sure thing for Player 1: Min($n-2, $n) = $n-2 < $98… i.e., is worse. So ($100, $100) is indeed a WFE

27. 3. CHOOSING AN EQUILIBRIUM BY VIRTUAL BARGAINING: REDUCING THE SET OF EQUILIBRIA

28. So coordination is not mere cooperation; and nothing to do with altruism How do we coordinate on the same equilibrium? • Virtual bargaining asks... • What would we agree? • And we all follow this agreement • If • there is an obvious winning agreement • Then • coordination can be achieved

29. The link between virtual bargaining and ‘real’ bargaining is a general psychological claim • Whatever factors influence ‘real’ bargaining should influence virtual bargaining • Personality • Reputation • Past history • Background wealth • Status, etc, etc

30. Formal challenge: choose a specific model of bargaining • E.g., Nash bargaining (not related to Nash equilibrium) • Maximize product of utility gains for the bargain, in comparison with some default • Not always obvious what the default should be… • And, assuming cautious players, we’ll consider the utility of the “sure thing” outcome • the worst case scenario

31. Some easy cases 1. Boobytrap game • Mix of C/D gives payoff of (29+, 29+) • Nash (D, D) gives 20, 20 • So we’ll agree to play the mixed strategy (even though it is dominated) • Of course, the ‘other’ may defect – but I’ve taken that into account

32. Experimental result(Misyak & Chater, in press) And people play a B, C mix; rarely D; And do better than Nash: D, D And no reliance on altruism, common interests Even antagonistic players will ‘virtually bargain’

33. Another easy case 2. Traveller’s Dilemma • All ($n, $n) are WFE • Sure thing payoffs ($n-2, $n-2) • Maximize this by making n as large as possible: i.e., choose ($100, $100)

34. And yet another easy case 3. Asymmetrical battle of the sexes • Simplifying – assume payoffs are utilities, and default is (0, 0) (Ballet, Ballet) product of gains is: (1-0)(9-0) = 9 (Football, Football): (10-0)(8-0) = 80 • Choose (Football, Football)

35. But not all cases will be ‘easy’ • Recall: General claim is that the challenges for virtual bargaining will be the same as for a theory of bargaining • And we can test this experimentally • (comparing cases where players can or can’t communicate) (Caveat: issue of the ‘veil of ignorance’ in symmetric coordination games)

36. 4. COMMUNICATION AS VIRTUAL BARGAINING Implication: Real agreements are built on virtual bargains

37. Individualist Theory of Action • Chosen to maximize the utility of the actor • The informational bankruptcy of “cheap talk” (Farrell, J.; Rabin, M. (1996). "Cheap Talk". Journal of Economic Perspectives10 (3): 103–118) • So signalling via action must be costly : e.g., “stotting” http://en.wikipedia.org/wiki/Stotting

38. Individualist theories (e.g., in economics) think ‘talk is cheap’ • And not to be trusted! • But then how does • Language • Facial expression • Gesture • … • Successfully carry information between people?? • Why aren’t we limited to: = “I’m fast!”

39. Joint action is ubiquitous in human activity—including organizations If it is obvious what we’d agree, we just do it! • (given common objective and common knowledge)

40. Virtual agreement sees communication as joint action - offers a way out… helps us jointly choose between (infinitely many!) different possible mappings “The cup…” ? Holmes What is ‘utility’? – something like: communicative content/cognitive effort Watson

41. “The cup…” implies a unique cup in the scene … so… = “cup” “cup” “trophy” = “cup” “cup” “trophy” = “cup” “mug” “cup” = “cup” “cup” “trophy”

42. That this is the ‘best’ mapping is common knowledge to Holmes and Watson “trophy” “The cup…” “trophy” “cup” Holmes Watson “trophy” So they both know the meaning of “the cup…” in this context; and can hence use the phrase to communicate successfully

43. Communication has “symmetrical” pay-offs... And miscommunication is bad for both of us Only worth sending a message if Watson understands it No point making an inference, if its not what Holmes meant So communication is a coordination ‘game’—like Hi-Lo

44. And does not presuppose cooperation No “Cooperative Principle” (e.g., H. P. Grice, 1975) Only worth sending a message if Watson understands it No point making an inference, if its not what Holmes meant

45. A simple linguistic example • “The red square”

46. “The red square” Reference determined by highly local “focal” convention

47. In coordination problems, past virtual bargains may be ‘focal points’ for future bargains • If, at t0, ‘cup’ = then, other things being equal, • At t1, ‘cup’ = not Potential for ‘layering’ of linguistic and other conventions

48. 5. APPLICATIONS AND FUTURE DIRECTIONS

49. Possible extension to dynamic games • Ultimatum game • Player 1 splits $100 (in integer units) • Player 2 then ‘accepts’ or ‘rejects’ (then both get nothing) • (rejecting is a bit like buying the Boobytrap---and is a dominated option) • Under simplifying assumptions, $50, $50 ‘wins’ (maximizes the product of utility gains)

50. The ability to ‘boobytrap,’ even though this option is ‘dominated,’ may be crucial… • …even to understand the most basic economic transaction • The unobserved one-off low-value exchange (e.g., buying a newspaper, cf. Bob Sugden) • ‘Punishing’ defection is costly— • But, by virtual bargaining, we can ‘precommit’ to it, even though precommitment is a dominated option