1. 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

4. ► Volunteer‘s Dilemma (VOD)

5. 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: 1 2 …
N - 1
Other C-Players C
U - K D 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.:
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)

7. 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

8. 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

9. 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 addresses from 41 universities
in Austria, Germany, and Switzerland

10. Results Group Size

11. ,
Gender, and Cost-Effect

12. 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!

13. 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”)

14. Design Experiment II 1. Symmetric VOD (control): U = 80, K = 50
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.

15. Results Experiment II: The Importance of Learning

16. Results Experiment II: Dynamics of Behaviour

17. Results Experiment II: Exploitation of the strong player

18. 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.

19. Solution of a „Volunteer‘s Dilemma“ by Emperor Penguins
Wikipedia Commons

21. 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

22. Thomas Feldman
Sarah Feldman

23. Experiment I: „E-Mail help requests“
Replication and extension of
Barron and Yechiam (2002)
Group size 1, 2, 3, 4, 5
Core hypothesis: Decline
of cooperation with group size N

27. Design Experiment II: Asymmetric VOD

29. 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.

30. Design Experiment II: Symmetric VOD

32. Results Experiment II

33. Results Experiment II

34. Results Experiment III

35. Results Experiment III

36. Results Experiment III