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Multilateral Bargaining

Game Description

A group of zombies has a number of humans to divide amongst themselves. Each round, each zombie proposes a division of the current stock of humans. One of the proposals is chosen at random and voted upon. If a majority approve, the humans are divided according to the proposal. If not, a fraction of the humans escape, and we move to the next round. Rounds progress until a proposal is accepted, and a player’s payoff is the number of humans he receives in the approved proposal.

Learning Objective 1: Coalition Formation Under Majority Rule

This game teaches students about the power to propose, and the importance of organizing coalitions. Students explore how majority rule may lead to unequal distribution of resources.

Learning Objective 2: Solving with Backward Induction

In more advanced classes, this game may be used as an example of backward induction. In particular, once we establish a player’s expected payoff if the game proceeds to the next round, we know the minimum acceptable proposal this round.

Brief Instructions

The default parameters clearly highlight the two main predicted outcomes of coalition formation under majority rule: relatively high payoffs for the proposer and unequal treatment of those whose votes are not needed for proposal passage.

By default, Group Size=3 zombies start with Total Pie=25 humans, and between each round Discount Factor %=10 percent of the humans escape. In the subgame-perfect equilibrium, each player should, in round one, propose 17 humans for herself and 8 for one of the other 2 players.

In general, most students learn quickly to form coalitions, make proposals that give nothing to those whose votes are not needed. Thus while the first iteration may last multiple rounds for some groups, subsequent iterations tend to proceed rather quickly.

Note that if you set Periods>1, group composition (and player identities) remain fixed across rounds. To reduce repeat-game effects (and spread learning across groups more quickly) set Periods=1. After finishing a one-period game, choose Replay and then Random to have group assignments randomized between games.

Finally, while the game ends for a group when a majority accept the randomly selected proposal, it will also end when there is only one human left or # of Rounds have been played without a proposal accepted. In these latter cases, each player earns zero.

Equilibrium Prediction

We start with the assertion that at any point on the equilibrium path, the game ends in the current round, and assume a player does not differentiate other players. If there will be X humans next period if the current proposal is rejected (off the equilibrium path), then by player symmetry the expected value of continuing to next period is X/(Group Size). Therefore a player accepts any proposal where she receives more than X/(Group Size) humans.

In equilibrium, each player’s round-one proposal is (Total Pie×(1–Discount Factor))/(Group Size) for half of the other players, and the rest for herself. With the default parameters and given then humans must be in whole units, there will be floor[25×(1–.1)]=22 humans in round 2 if the round-one proposal is rejected. As the expected value of continuing to round 2 is 22/3=71/3, each player proposes 8 humans for 1 of the other 2 players, and 17 for herself.

Key Treatment Variations

We have chosen the default parameters to highlight the 2 main results of coalition formation under majority rule:

By default, # of Rounds equals 100. In fact, when there is only one human remaining, it is allocated to one of the players and the game ends even if there are additional rounds.

Setting # of Rounds to a small number may make it easier to teach the intuition behind the game’s equilibrium. This is due to the fact that round-one equilibrium play is the same for any number of rounds greater than 1.

Finally, you should expect coalition formation to be more difficult as Group Size increases. Note that in practice—in order to make sure the coalition sticks together—proposers are generally more generous than predicted. As group size increases, this extra generosity can quickly eat away at the “power to propose”.

In any event, making sure that Group Size is odd makes the target coalition size more salient.

Results

Figure 1: Summary Table

We first present a table showing agreements that are made among groups (Figure 1). The row titled "Predicted" shows what the equilibrium allocation would be, for comparison. When students play, it is common to see a more equitable distribution of humans than theory predicts.

Figure 2: Proposal Summary

The next table summarizes rejected and accpeted proposals by coalition size, and Round 1 proposals (Figure 2).

Figure 3: Voted Proposals

We then present a graph showing proposals across rounds (Figure 3). Proposals which are rejected tend to favor one group member enough that coalition formation is not possible.

Figure 4: All Proposals

Finally, we present a graph showing an average of all proposals in each round (Figure 4). You can dynamically choose which elements to show in the graph. Click on the checkbox next to the item name in the graph legend to remove it. For example, you can choose to show only Proposer Allocation in the graph to compare how allocations change across rounds, as the total number of humans to split decreases.

If you have played a multi-period game, use the Go To: drop down menu to switch between periods. You can also select a Cross-Period summary from the Go To: menu, which presents a graph showing proposals across periods. If you have replayed a game, click on the results button in the last game run to compare results across games.

Robot Play

Our robot (i.e., an automated player) strategies for each role are the following:

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