MERITOCRACY AS A MECHANISM IN THE CONTEXT OF VOLUNTARY CONTRIBUTION GAMES

Transcription

1 MERITOCRACY AS A MECHANISM IN THE CONTEXT OF VOLUNTARY CONTRIBUTION GAMES HEINRICH H. NAX (HNAX@ETHZ.CH) COSS, ETH ZURICH MAY 26, 2015

2 BUT BEFORE WE BEGIN Let us clarify some basic ingredients of the course: 1. Game Theory 2. Social Preference Theory 3. Mechanism Design 4. Collective Intelligence

3 1. GAME THEORY A mathematical language to express models conflict and cooperation between intelligent rational decision-makers (Myerson) In other words, interactive decision theory (Aumann) Dates back to von Neumann & Morgenstern (1944) Most important solution concept: the Nash (1950) equilibrium

4 2. SOCIAL PREFERENCE THEORY Perhaps we should have defined game theory as interactive decision theory involving rational and SELFISH decision-makers SELFISH = self-regarding in a narrow sense Social preference allows for other concerns such as altruism fairness considerations reciprocity etc.

5 3. MECHANISM DESIGN Think of it as reverse game theory in a design problem, the goal function is the main given, while the mechanism is the unknown. (Hurwicz) The mechanism designer is a game designer. He studies What agents would do in various games And what game leads to the outcomes that are most desirable

6 4. COLLECTIVE INTELLIGENCE Collective intelligence is intelligence that is shared by a group of interacting individuals Models of collective intelligence have been formulated for animal behavior and for human behavior Collective intelligence may be the emergent outcome of interactive collaboration or competition

7 MERITOCRACY AS A MECHANISM OR A SORT-OF INTRODUCTION TO THE DIFFERENT COMPONENTS OF THE COURSE

8 SOURCES joint work with other colleagues at ETH Zurich: Stefano Balietti Dirk Helbing Ryan Murphy two parts: theory experiments ( paper (w. DH+RM) available: 2 of 52

9 General message of this talk: TALK ABOUT MERITOCRACY & EFFICIENCY-EQUALITY T/OFF AS THE TARGET OF A SOCIAL PLANNER INTRODUCE THE MECHANISM OF MERITOCRACY STUDY THE CONTEXT OF VOLUNTARY CONTRIBUTION GAMES 3 of 52

10 MERITOCRACY : DEFINITION OF THE BASIC MECHANISTIC IDEA Def.: rule by those with merit/ rule rewarding merit old concept with a surprisingly new name (Young 1958) present in early modern societies including China, Greece, Rome examples include selection of officials/ councilmen, military reward/ promotion schemes and access to education proposed by thinkers such as Confucius, Aristotle and Plato Criticism: as identified, for example, in the book by Arrow, Bowles and Durlauf (2000) is the inherent inequality-efficiency trade-off (e.g. education) 4 of 52

11 SIMPLE VOLUNTARY CONTRIBUTION GAME every player ii chooses whether to contribute (cc ii = 11) or not (cc ii = 00) given contributions, players are matched (how remains to be specified) into groups of fixed size s given contributions in each group GG, for a marginal per-capita rate of return (mmmmmmmm) rr/ss (11/ss, 11), a public good is provided and its return split equally so that ii s payoff is uu ii (cc) = 1 cc ii + mmpppppp cc jj jj GG ii : ii GG ii 5 of 52

12 BEST REPLIES; ONE GROUP Suppose nn = 44, ss = 44, mmmmmmmm = ; i.e. there is only one group DILEMMA: THE ONLY EQUILIBRIUM IS ALL FREE-RIDE = worst efficiency! (Upside: The outcome is perfectly equal.)

13 BEST REPLIES YOU OTHERS CONTRIBUTE (1) 1 payoff 2 1 payoff 2 1 payoff 2 1 payoff 2 FREE-RIDE (0) 0 payoff payoff payoff payoff 1.5 YOU OTHERS 1 payoff payoff payoff payoff payoff 1 0 payoff 1 0 payoff 1 0 payoff 1

14 FREE-RIDING IS A DOMINANT STRATEGY IF that is, uu ii cc cc ii < 00 if: For example: nn = ss, i.e. there is only one group (Marwell and Ames 1979) or groups are randomly matched (Andreoni 1988) (Upside: in a homogenous population, the outcome is perfectly equal.) 6 of 52

15 WHAT HAPPENS UNDER MERITOCRATIC MATCHING 1. actual contributions cc ii are chosen by players 2. Gaussian noise with mean 0 and variance 11/ββ added to actual contributions 3. ββ is the index of meritocracy in the system 4. players are ranked by noised contributions 5. the payoffs materialize based on actual contributions 7 of 52

16 ββ-meritocratic MATCHING ββ 00 ββ No meritocracy Intermediate level of meritocracy Perfect meritocracy 8 of 52

17 Specific message of this talk: ANALYZE: MERITOCRATIC MATCHING IN VOLUNTARY CONTRIBUTION GAMES WHERE THE KEY ISSUE IS NOT DISTRIBUTIONAL (AS WITH A FIXED RESOURCE) INSTEAD ABOUT THE CREATION OF A RESOURCE KEY QUESTION: WHAT IS THE MINIMAL LEVEL OF MERITOCRATIC MATCHING FIDELITY TO INDUCE COOPERATION BY PURELY SELF-INTERESTED AGENTS? 9 of 52

18 LET S EXPLORE THE RANGE OF REGIMES 10 of 52

19 NON-MERITOCRACY (here: ββ 00) random one group: standard game (Marwell and Ames 1979, Isaac et al. 1985) with group matching: random (re-)matching (Andreoni 1988) Outcome: the only equilibrium is all free-ride (homogeneous population: max equality but min efficiency) 11 of 52

20 FULL MERITOCRACY (here: ββ ): perfect group-based mechanism: grouping by rank (Gunnthorsdottir et al. 2010) Outcome: if the mmmmmmmm is high enough, a new asymmetric equilibrium in pure strategies emerges where the majority contributes and a small minority free-rides (Gunnthorsdottir et al. 2010, Theorem 1). (homogeneous population: efficiency gain comes with inequality) 12 of 52

21 ENTIRE RANGE: (00 ββ 11) (see NAX ET AL. 2014, PROPOSITIONS 6-10) If mmmmmmmm/ββ high enough, there are new Nash equilibria: n players free-ride m<n players contribute every player contributes with probability p>0 MPCR 13 of 52 MERITOCRACY

22 SOME BEST REPLY EXAMPLES; PERFECT MERITOCRACY Suppose nn = 88, ss = 44, mmmmmmmm = , ββ ; i.e. two groups are matched under perfect merticracy EXAMPLE 1: Others: all contribute 0 You: What is the best reply? 14 of 52

23 EXAMPLE 1: BEST REPLY CONTRIBUTE (1) 1 payoff FREE-RIDE (0) payoff of 52

24 EXAMPLE 2: Others: all contribute 1 You: What is the best reply? 16 of 52

25 EXAMPLE 2: BEST REPLY CONTRIBUTE (1) payoff FREE-RIDE (0) payoff of 52

26 EXAMPLE 3: Others: You: 4 contribute 0 What is the best reply 3 contribute 1 18 of 52

27 EXAMPLE 3: BEST REPLY CONTRIBUTE (1) payoff of 52 FREE-RIDE (0) payoff payoff p=0.2 p=0.8 EXPECTATION (0) = 1.3<2

28 EVOLUTIONARY DYNAMICS suppose the game is repeated at time steps t=1,2,3, 20 of 52

29 PERTURBED BEST REPLY The same n agents Each period, each agent plays a myopic best reply to the previousperiod actions of the n-1 other players and the other action with probability 1-ε with probability ε (we can think of ε as a myopic best reply error rate here) 21 of 52

30 STOCHASTIC STABILITY Definition: A state is stochastically stable (Foster and Young 1990) if the stationary distribution as ε goes toward 0 places positive weight on that state. 22 of 52

31 theoretical results STOCHASTIC STABILITY (see NAX ET AL. 2014, LEMMA 3) CONTRIBUTION 23 of 52 RANDOM ββ sssssssss_ssssssssssss MERITOCRACY PERFECT

32 Now let us think about the WELFARE of equilibria Say social welfare given inequality aversion parameter ee 00, is 11 WW ee uu = nn 11 ee uu ii 11 ee ii NN (When ee = 11, assume WW ee uu = 11 nn ii NN uu ii, i.e. the Nash product.) WW ee uu is a variant of the function by Atkinson (1970) nesting Benthiam welfare if ee = 00 (straight-up Utilitarianism) Rawlsian welfare if ee (extreme inequality aversion) 24 of 52

33 WELFARE ILLUSTRATIONS No inequality aversion Intermediate inequality aversion Extreme inequality aversion WW ee uu with ee < PERFECT MERITOCRACY EQUILIBRIUM WW ee uu with ee > NON-MERITOCRACY EQUILIBRIUM 25 of 52

34 MERITOCRATIC MATCHING: EXPERIMENTS 26 of 52 34

35 Experiments: SET-UP i/ii Experiments were conducted in May/ June ETH Zurich s DeSciL (involving 192 subjects in 12 sessions) In each session, 16 players played two of our games The mmmmmmmm was always 0.5 and the group size always 4 The budget was 20 coins each round The game was repeated 40 rounds Players received full instructions and (anonymous) feedback about previous-period play Play was incentivized with real money (one coin=0.01 CHF) Average earning were 40 CHF (for one hour experiment) Games differed w.r.t. variance levels: 00, 33, 2222, oooo. (Note: when, the near-efficient equilibrium does not exist.) 27 of 52

36 Experiments: SET-UP ii/ii Each player played two games; each one with a different variance level All possible ordered pairs of variance levels were played and made up a separate session A somewhat hybrid design: between-subject/ within-subject Each player experienced either a meritocracy increase or a meritocracy decrease 12 sessions = 6 possible variance pairs * 2 orders each 28 of 52

37 Screenshots 29 of 52

38 Screenshots 30 of 52

39 Screenshots 31 of 52

40 MATCHING UNDER RANDOM 32 of 52

41 MATCHING UNDER PERFECT 33 of 52

42 MATCHING UNDER VAR=3 34 of 52

43 EXPERIMENTAL EVIDENCE 35 of 52

44 CONTRIBUTIONS: PERFECT 36 of 52

45 CONTRIBUTIONS: Var3 37 of 52

46 CONTRIBUTIONS: Var20 38 of 52

47 CONTRIBUTIONS: RANDOM 39 of 52

48 AGGREGATE PATTERN: 40 of 52

49 EFFICIENCY 41 of 52

50 PAYOFF DISTRIBUTION: PERFECT 42 of 52

51 PAYOFF DISTRIBUTION: Var3 43 of 52

52 PAYOFF DISTRIBUTION: Var20 44 of 52

53 PAYOFF DISTRIBUTION: RANDOM 45 of 52

54 (IN)EQUALITY (VAR AND GINI) 46 of 52

55 FAIRNESS CONSIDERATIONS note to myself: compare with Fehr-Schmidt/B-O Fairness leads to contribution decrease leads to contribution increase 47 of 52

56 FAIRNESS 48 of 52

57 FAIRNESS 49 of 52

58 FAIRNESS (F-S DOES NOT WORK) Question: Have we proposed a fairness concept that is not a social preference? 50 of 52

59 CONCLUSION Moderate levels of meritocracy enable new, nearefficient equilibria New equilibria are more stable if the mechanism is sufficiently meritocratic New equilibria are typically preferable w.r.t. social welfare even with substantial inequality aversion Efficiency-inequality tradeoff is moderated In practice, a hint of meritocracy may prove sufficient to reach more efficient outcomes with high contributions Realized inequality is lower in higher meritocracy regimes Efficiency-inequality tradeoff dissolved THEORY EXPERIMENTAL LESSONS EVIDENCE 51 of 52

60 REFERENCES 52 of 52