Special case: a "strong" cooperative player receives Uk Kk, n-1 symmetric "weak" players’ get u k whereby

• 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!

• 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”)

Group size (symmetric game, experiment III, not presented):

• 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

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

Choose the probability of defection q such that an actor is indifferent concerning the outcome of strategy C and D

• 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

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.

• 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.