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Experiments with the Volunteer‘s Dilemma Game. Andreas Diekmann, Wojtek Przepiorka, Stefan Wehrli ETH Zurich Presentation for the Social Dilemma Conference, Kyoto, August 20-24, 2009. ► Volunteer‘s Dilemma (VOD). Volunteer’s Dilemma
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Experiments with the Volunteer‘s Dilemma Game Andreas Diekmann, Wojtek Przepiorka, Stefan Wehrli ETH Zurich Presentation for the Social Dilemma Conference, Kyoto, August 20-24, 2009
Volunteer’s Dilemma The Volunteer’s Dilemma is a N-player binary choice game (for N ≥ 2) with a step-level production function. A player can produce a collective good at cost K > 0. When the good is produced, each player obtains the benefit U > K > 0. If no player volunteers, the good is not produced and all players receive a payoff 0. N-Player Game: → No dominant strategy, but N asymmetric pure equilibria in which one player volunteers while all other defect. Cf. Diekmann (1985) Formal Models & Extensions,e.g.: • Volunteer’s Dilemma (Diekmann 1985) • Asymmetric VOD (Diekmann 1993, Weesie 1993) • Timing and incomplete information in the VOD (Weesie 1993, 1994) • Cost sharing in the VOD (Weesie and Franzen 1998) • Evolutionary Analysis of the VOD (Myatt and Wallace 2008) • Dynamic Volunteer’s Dilemma (Otsubo and Rapoport 2008)
Mixed Nash Equilibrium Choose the probability of defection q such that an actor is indifferent concerning the outcome of strategy C and D U – K = U (1 – qN-1) U – K = U - U qN-1 qN-1 =K/U Defection ► q = N-1√ K/U Cooperation ► p = 1 - N-1√ K/U “Diffusion of Responsibility” derived from a game model
Experiment 1 „E-Mail Help Request“ Replication and extension of Barron and Yechiam (2002) Hello! I am a biologist student and I want to continue my studies at the University of (...). Can you please inform me if there is a biology department at the university of (...)? Thank you very much for your help. Sarah Feldman Low cost help, K is low
Treatments and Sample High(er) cost help, K is high Sarah pretends to have to write an essay in school on „Human beings and apes“. She regrets that her school has no online access to the literature and she asks to send to her a pdf of an article in „Nature“ by F.D. Waals on „Monkeys reject unequal pay“. • Group size: N = 1, 2, 3, 4, 5 (Nash-equilibrium hyothesis = diffusion of responsibilty) • Low cost versus high(er) cost, increasing K • Personalized letter (Dear Mrs. Miller) • Gender: Sarah Feldman versus Thomas Feldman • (Asymmetric situation, not reported here) Sample: 2 samples of 3374 e-mail addresses from 41 universities in Austria, Germany, and Switzerland
, Gender, and Cost-Effect
Asymmetric Volunteer’s Dilemma Heterogeneity of costs and gains: Ui, Ki for i = 1, …, N; Ui > (Ui – Ki) > 0 Special case: A “strong” cooperative player receives Uk – Kk , N-1 symmetric “weak” players’ get U – K whereby: Uk – Kk > U – K Strategy profile of an “asymmetric”, efficient (Pareto optimal) Nash equilibrium: s = (Ck, D, D, D, D, …,D) i.e. the “strong player” is the volunteer (the player with the lowest cost and/or the highest gain). All other players defect. ► In the asymmetric dilemma: Exploitation of the strong player by the weak actors. ►Paradox of mixed Nash equilibrium: The strongest player has the smallest likelihood to take action!
Evolution of Social Norms in Repeated Games Asymmetric VOD. There are two types of Nash-equilibria in an asymmetric VOD. The mixed Nash-equilibrium and the asymmetric pure strategy equilibrium. We expect the asymmetric equilibrium to emerge in an asymmetric game. In our special version, we expect a higher probability of the “strong” player to take action, i.e. to cooperate than “weak” players. (Exploitation of the “strong” by the “weak”)
Design Experiment II Group size N = 3, series of interactions with 48 to 56 repetions, 120 subjects 1. Symmetric VOD (control): U = 80, K = 50 2. Asymmetric 1: Weak players: U = 80, K = 50, Strong player: U = 80, K = 30 3. Asymmetric 2: Weak players: U = 80, K = 50, Strong player U = 80, K = 10 The collective good has always the same value U = 80. Strong players have lower cost to produce the collective good.
Conclusions: Asymmetric Volunteer‘s Dilemma • Group size (symmetric game, experiment III, not presented): Complexity of coordination. The efficient provision of the public good decreases with group size. • Learning: Evolution of norms. The efficient provision of public good and successfull coordination is increasing with the number of repetitions of the game. • Player‘s strength: Exploitation of strong players. In the asymmetric VOD, the public good was provided to a much higher degree by the strong player compared to weak players (although the probability of efficient public good provision was not larger than for the symmetric VOD). Strong player‘s higher likelihood of action is expected by the Harsanyi/Selten theory of equilibrium selection.
Wikipedia Commons Solution of a „Volunteer‘s Dilemma“ by Emperor Penguins
Mixed Nash Equilibrium Choose the probability of defection q such that an actor is indifferent concerning the outcome of strategy C and D U – K = U (1 – qN-1) U – K = U - U qN-1 qN-1 = K/U Defection ► q = N-1√ K/U Cooperation ► p = 1 - N-1√ K/U
Thomas Feldman Sarah Feldman
Experiment I: „E-Mail helprequests“ Replication and extension of Barron and Yechiam (2002) Group size 1, 2, 3, 4, 5 Core hypothesis: Decline of cooperation with group size N
Experiments II and III: Evolution of Social Norms in Repeated Games: Hypotheses H1 Repeated symmetric VOD. There are N asymmetric Nash-equilibria in a stage game. Actors take turn to volunteer (C). The outcome is “efficient” (Pareto optimal) and actors achieve an equal payoff distribution with payoff U – K/N per player. a) We expect a coordination rule to emerge over time (social learning) b) Coordination is more problematic in larger groups. H2 Asymmetric VOD. There are two types of Nash-equilibria in an asymmetric VOD. The mixed Nash-equilibrium and the asymmetric pure strategy equilibrium. a) We expect the asymmetric equilibrium to emerge in an asymmetric game. In our special version, we expect a higher probability of the “strong” player to take action, i.e. to cooperate than “weak” players. (Exploitation of the “strong” by the “weak”) b) A focal point should help to solve the coordination problem and the focal player is expected to develop a higher intensity of volunteering.
