STABILITY AND WELFARE OF MERITOCRATIC GROUP-BASED MECHANISMS

Transcription

1 STABILITY AND WELFARE OF MERITOCRATIC GROUP-BASED MECHANISMS HEINRICH H. NAX MAY 5, 25 SNI MONTE VERITA

2 SOURCES joint work with Stefano Balietti Dirk Helbing Ryan Murphy two parts: TODAY: theory YESTERDAY: experiments paper (w. DH+RM) available: 2 of 25

3 General message of this talk: TALK ABOUT MERITOCRACY & EFFICIENCY-EQUALITY T/OFF INTERPRET ASSORTATIVITY= MERITOCRATIC MATCHING STUDY THE CONTEXT OF VOLUNTARY CONTRIBUTION GAMES 3 of 25

4 MERITOCRACY : DEFINITIONS Def.: rule by those with merit/ rule rewarding merit old concept with a surprisingly new name (Young 958) 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 (2) is the inherent inequality-efficiency trade-off (e.g. education) 4 of 25

5 SIMPLE VOLUNTARY CONTRIBUTION GAME every player ii chooses whether to contribute (cc ii = ) or not (cc ii = ) 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 (/ss, ), a public good is provided and its return split equally so that ii s payoff is uu ii (cc) = cc ii + mmpppppp cc jj jj GG ii : ii GG ii 5 of 25

6 FREE-RIDING IS A DOMINANT STRATEGY IF that is, uu ii cc cc ii < if: For example: nn = ss, i.e. there is only one group (Marwell and Ames 979) or groups are randomly matched (Andreoni 988) (Upside: in a homogenous population, the outcome is perfectly equal.) 6 of 25

7 WHAT HAPPENS UNDER MERITOCRATIC MATCHING. actual contributions cc ii are chosen by players 2. Gaussian noise with mean and variance /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 25

8 ββ-meritocratic MATCHING ββ ββ No meritocracy Intermediate level of meritocracy Perfect meritocracy 8 of 25

9 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 25

10 LET S EXPLORE THE RANGE OF REGIMES of 25

11 NON-MERITOCRACY (here: ββ ) random one group: standard game (Marwell and Ames 979, Isaac et al. 985) with group matching: random (re-)matching (Andreoni 988) Outcome: the only equilibrium is all free-ride (homogeneous population: max equality but min efficiency) of 25

12 FULL MERITOCRACY (here: ββ ): perfect group-based mechanism: grouping by rank (Gunnthorsdottir et al. 2) 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. 2, Theorem ). (homogeneous population: efficiency gain comes with inequality) 2 of 25

13 ENTIRE RANGE: ( ββ ) (see NAX ET AL. 24, PROPOSITIONS 6-) If mmmmmmmm/ββ high enough, there are new Nash equilibria: n players free-ride m<n players contribute every player contributes with probability p> MPCR 3 of 25 MERITOCRACY

14 SOME BEST REPLY EXAMPLES; PERFECT MERITOCRACY Suppose nn = 88, ss = 44, mmmmmmmm =. 55, ββ ; i.e. two groups are matched under perfect merticracy EXAMPLE : Others: all contribute You: What is the best reply? 4 of 25

15 EXAMPLE : BEST REPLY CONTRIBUTE () payoff.5 FREE-RIDE () payoff 5 of 25

16 EXAMPLE 2: Others: all contribute You: What is the best reply? 6 of 25

17 EXAMPLE 2: BEST REPLY CONTRIBUTE () payoff 2 FREE-RIDE () payoff of 25

18 EXAMPLE 3: Others: You: 4 contribute What is the best reply 3 contribute 8 of 25

19 EXAMPLE 3: BEST REPLY CONTRIBUTE () payoff 2 9 of 25 FREE-RIDE () payoff 2.5 payoff p=.2 p=.8 EXPECTATION () =.3<2

20 theoretical results STOCHASTIC STABILITY (see NAX ET AL. 24, LEMMA 3) CONTRIBUTION 2 of 25 RANDOM ββ sssssssss_ssssssssssss MERITOCRACY PERFECT

21 Now let us think about the WELFARE of equilibria Say social welfare given inequality aversion parameter ee, is WW ee uu = nn ee uu ii ee ii NN (When ee =, assume WW ee uu = nn ii NN uu ii, i.e. the Nash product.) WW ee uu is a variant of the function by Atkinson (97) nesting Benthiam welfare if ee = (straight-up Utilitarianism) Rawlsian welfare if ee (extreme inequality aversion) 2 of 25

22 WELFARE ILLUSTRATIONS No inequality aversion Intermediate inequality aversion Extreme inequality aversion WW ee uu with ee <. 33 PERFECT MERITOCRACY EQUILIBRIUM WW ee uu with ee >. 33 NON-MERITOCRACY EQUILIBRIUM 22 of 25

23 theoretical results WELFARE (see NAX ET AL. 24) Result: social planner setting ββ will set ββ = (no meritocracy) only if very inequality averse, else he sets ββ = ββ ssssssssss_ssssssssssss (intermediate meritocracy). Reason: depending on his inequality aversion ee, the nearefficient pure-strategy Nash equilibrium or the free-riding equilibrium is preferred (efficiency-inequality trade-off). 23 of 25

24 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 THEORY LESSONS 24 of 25

25 REFERENCES 25 of 25