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
The Volunteer’s Dilemma is a N-player binary choice game (for N ≥ 2) with a steplevel 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:
0
1
2
…
N-1
C
U-K
U-K
U-K
…
U-K
D
0
U
U
…
U
Other C-Players
→ 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, Sarah pretends to have to write an essay
K is high
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 K
3. Personalized letter (Dear Mrs. Miller)
4. Gender: Sarah Feldman versus Thomas Feldman
5. (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
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.
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 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 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
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