Experiments with the Volunteer‘s Dilemma Game

Andreas Diekmann, Wojtek Przepiorka, Stefan WehrliETH Zurich

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

► Volunteer‘s Dilemma (VOD)

Volunteer’s DilemmaThe 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:

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)

0 1 2 … N - 1 Other C-Players

C U - K U - K U - K … U - K

D 0 U U … U

→ 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.:

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

Mixed Nash Equilibrium

Experiment 1 „E-Mail Help Request“Replication 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 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“.

1. Group size: N = 1, 2, 3, 4, 5 (Nash-equilibrium hyothesis = diffusion of responsibilty)

2. Low cost versus high(er) cost, increasing K3. Personalized letter (Dear Mrs. Miller)4. Gender: Sarah Feldman versus Thomas Feldman5. (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 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.

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.

• 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.

Conclusions: Asymmetric Volunteer‘s Dilemma

Solution of a „Volunteer‘s Dilemma“ by Emperor Penguins

Wikipedia Commons

Mixed Nash Equilibrium

Choose 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-1

qN-1 = K/UDefection q = ► N-1√ K/U

Cooperation p = 1 - ► N-1√ K/U

Sarah Feldman

Thomas Feldman

Experiment I: „E-Mail help requests“

Replication 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: 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.

Design Experiment II: Symmetric VOD

Results Experiment II

Results Experiment II

Results Experiment III

Results Experiment III

Results Experiment III