Experiments with the Volunteers Dilemma GameAndreas Diekmann, Wojtek Przepiorka, Stefan WehrliETH Zurich

Presentation for the Social Dilemma Conference, Kyoto,August 20-24, 2009

Volunteers Dilemma (VOD)

Volunteers DilemmaThe Volunteers 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:

Volunteers 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 Volunteers Dilemma (Otsubo and Rapoport 2008)

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.:

012N - 1Other C-PlayersCU - KU - KU - KU - KD0UUU

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/UDiffusion of Responsibilityderived from a game model

Mixed Nash Equilibrium

Experiment 1 E-Mail Help RequestReplication and extension ofBarron 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 biologydepartment at the university of (...)?

Thank you very much for your help.

Sarah Feldman Low cost help,K is low

Treatments and SampleHigh(er) cost help,K is highSarah 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 KPersonalized 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

Results Group Size

Gender, and Cost-Effect,

Asymmetric Volunteers 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 IIGroup 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.

Results Experiment II: The Importance of Learning

Results Experiment II: Dynamics of Behaviour

Results Experiment II: Exploitation of the strong player

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.

Players 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 players higher likelihood of action is expected by the Harsanyi/Selten theory of equilibrium selection.

Conclusions: Asymmetric Volunteers Dilemma

Solution of a Volunteers Dilemma by Emperor PenguinsWikipedia Commons

Mixed Nash EquilibriumChoose the probability of defection q such that an actor is indifferent concerning the outcome of strategy C and DU K = U (1 qN-1)U K = U - U qN-1qN-1 = K/UDefection q = N-1 K/UCooperation p = 1 - N-1 K/U

Sarah FeldmanThomas Feldman

Experiment I: E-Mail help requestsReplication and extension ofBarron and Yechiam (2002)Group size 1, 2, 3, 4, 5

Core hypothesis: Declineof cooperation with group size N

Design Experiment II: Asymmetric VOD

Experiments II and III: Evolution of Social Norms in Repeated Games: HypothesesH1 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.

Design Experiment II: Symmetric VOD

Results Experiment II

Results Experiment II

Results Experiment III

Results Experiment III

Results Experiment III