Request Game

The request game is structured different than the investment game. Within the investment game an individual's return would always be positive as long as they retained all their tokens and did not invest in the investment account. This implies that a subject could determine their payoff with certainty under this circumstance and the externalities assocaited with the traditional CPR would not be present. Within the request game this is not possible.

Within the request game all subjects are informed that they are to decide how many tokens they wish to request from a "pot" with a value uniformly distributed between a and b -- subjects know the distribution, but not the exact value of the size of the pot.

An individual's return is determined by the total requests made by the group and the realized value of the "pot." If the sum of the requests is less than or equal to the realized value, all will receive their request. If the aggregate request is greater than the realized value of the "pot" then everyone receives 0 tokens. Therefore, if everyone keeps their individual requests low, it is very probable that they will receive their request. However, if you believe that everyone is keeping their request low than you have an incentive to increase your request and earn more than the others. This introduces the "tragedy of the commons."

What makes this game interesting & relevant is the idea, first devised by Messick et al, of reducing the multi-stage replenishable resource trap to a single-stage task. This eliminates inter-trial dependencies, but introduces environmental uncertainty, and introduces a strong element of realism.

Nash Equilibrium

For the simultaneous contribution game, where a and b are the lower and upper bounds of the distribution, and there are n risk-neutral players, the Nash equilibrium strategy for a player is to request:
a/n if n*b < (n + 1)*a, and
b/(n + 1) if n*b (n + 1)*a
As we can see, each player's request first decreases and then increases as the range (b-a) of the distribution function increases.

However, it is far more complex for a sequential game. The request for the first player in the sequence follows the above pattern, i.e. it first decreases and then increases in the range b-a, while for the other n-1 players, the predicted request increases as b-a increases.

The Experiments

• Simultaneous game:
  The game was played with 80 subjects, divided into 16 groups so that n = 5. Three distributions were used: a = b = 500; a = 250, b = 750; and a = 0, b = 1000. Afterwards, in half the trials, subjects were asked to guess the size of the "pot" and the requests of the other players in their group.
  The average amount requested in the first case was 109, with a total mean request of 546. The good was provided 47% of the time, so the provision rate was 0.47. In the second case, the average amount requested was 122, the total mean request was 611 and the provision rate was 0.39, or 39%. In the third case, the average amount requested was 162, the total mean request was 811, and the good was provided 11% of the time, so the provision rate was 0.11. The mean individual request across all treatments was 131.
  
• Sequential game:
  This time, there were 45 subjects divided into 9 groups. The distributions were identical to those in the simultaneous game. Each of the three conditions above were replicated 10 times, so that each subject was assigned to the same position twice, though on random occasions.
  In this case, uncertainty initially decreased the request amount for the first three players, so that their mean requests in the a = b = 500 treatment was greater than the mean request in the a = 250, b = 750 treatment. This was reversed for the last two players, so their requests were greater in the a = 250, b = 750 treatment -- presumably they realized they could afford to request more. Player 3's mean request of 113 in the a = 250, b = 750 treatment was only marginally less than player 3's mean request of 114 in the a = b = 500 treatment. Each player's request was significantly higher in the a = 0, b = 1000 treatment.
  The resource provision rate was much higher in the sequential game. In the a = b = 500 treatment, the total mean request was 565 with a 64%, or 0.64 provision rate. In the a = 250, b = 750 treatment, the total mean request was 585, with a provision rate of 50%, or 0.5. In the a = 0, b = 1000 treatment, the total mean request was 765, with a provision rate of 23% or 0.23.
  

Findings

This game was created to analyze how the range of uncertainty in the resource, measured by the difference between beta and alpha, and the ordering of subjects effects subject behavior. Within this game Rapoport and Suleiman discovered that as the range of uncertainty increased so did the individual requests, with requests being focused on the equal split instead of the Nash equilibrium strategy, (b - a)/n, when the resource level was known with certainty. In all treatments, the equilibrium model overestimates the actual requests, while the equal share model underpredicts them.

In addition, to this finding they discovered that individual requests would decline in the order in which they were received with the last individual requesting the least from the resource when the game was structured as a sequential move game instead of a simultaneous move game. The first few subjects to choose did not take advantage of their position to the extent that the equilibrium strategy predicts; however, that calculation assumed risk-neutrality, while subjects can safely be assumed to be risk-averse.

References

1. Budescu, David V., Rapoport, Amnon, and Suleiman, Ramzi. Simultaneous vs. sequential requests in resource dilemmas with incomplete information. Acta Psychologica 80 (1992), 297-310.
2. Messick, D.M., S.T. Allison, & C.D. Samuelson. (1988) Framing and Communication Effects on group members' responses to environmental & social uncertainty. In S. Maital (Ed.), Applied behavioural economics, Vol. 2 (pp. 677-700). New York: New York University Press.
3. Rapoport, A., Budescu, D. V., & Suleiman, R. (1993). Sequential requests from randomly distributed shared resources. Journal of Mathematical Psychology, 37, 241-265.
4. Suleiman, R., & Rapoport, A. (1988). Environmental and social uncertainty in single trial resource dilemmas. Acta Psychologica, 68, 99-112.