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Bargaining: Alternating Offer

Game Description

This game is a multi-round version of the Ultimatum game. Pairs of students decide on how to split a pot of money by alternating who makes the accept-or-reject proposal. If rejected, the amount of money shrinks, each student receives zero, and the pair proceeds to the next round. If a proposal is agreed to, the students receive the agreed upon split.

Learning Objective 1: Tradeoffs and Fairness in Negotiations

Features of a two-party negotiation may offer an advantage to a particular party (e.g. information asymmetry). However, fairness considerations, which are an important component of two-party negotiations, may lessen any advantage.

Learning Objective 2: Solving with Backward Induction

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

Learning Objective 3: Equilibrium in the Indefinitely Repeated Game

In more advanced classes, this game may be used to illustrate subgame-perfect Nash equilibria in indefinitely repeated games. In this case, first-mover advantage is increasing as players become more impatient (because the pie shrinks more rapidly between rounds).

Brief Instructions

Rounds progress until players have reached an agreement, there are no further rounds, or until the amount available to be split goes to zero. In the first case, the accepted proposal determines player payoffs. In the latter two cases, each player earns zero.

As discussed below, the round 1 proposal (\(\text{keep}_1^*; \text{Total Pie}-\text{keep}_1^*\)) is accepted in equilibrium. Intuitively, the equilibrium round 1 proposal depends on the balance of two advantages: a first (or current) mover advantage and a last-mover advantage. The parameters you set for a game determine the relative size of these advantages, and therefore the degree of inequality in the equilibrium outcome.

With default parameters, the game starts with Total Pie=100 dollars, which shrinks by Shrink Rate=10 percent each time a proposal is rejected, and the game lasts no more than 9 Rounds ( Minimum # of Rounds=9 and Ending Probability (%)=100 percent). With an odd number of rounds, player 1 has both the first and last mover advantage, although with more than a few rounds, most students will overlook last-mover advantage. In the Nash equilibrium, \(\text{keep}_1^*=74\).1 Player 2 rejection of equilibrium-like proposals is consistent with fairness considerations, and more generous round 1 offers are consistent both with player 1’s preference for fairness and player 1 anticipating player 2’s preference for fairness.

Key Treatment Variations

To highlight backward induction, limit the finitely repeated game to only a few rounds (most humans engage in few rounds of backward induction). For example, vary Minimum # of Rounds from 2 to 3. Under the assumption that players accept proposals when indifferent, proposers \(keep^*_1=10\) in the two-round game. This amount jumps to 91 in the three-round game.

In general, most students learn the power of making the last proposal after a few iterations. While the first iteration may last multiple rounds for some groups, subsequent iterations tend to proceed rather quickly.

Note: 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 both group assignments and player roles randomized between games.

To implement the indefinitely repeated game, set Minimum # of Rounds=1 and Ending Probability (%)<100. As discussed below, the first-mover advantage (and thus predicted outcome inequality) increases as your increase either Ending Probability (%), or the rate at which the pie shrinks between rounds (Shrink Rate (%)).

Results

In general, you should expect to see an increase in round 1 agreements as students gain experience. When unequal splits are predicted, actual agreements will tend to be more equal, consistent with fairness considerations. If you have played a multi-period game, use the Go To dropdown menu to switch between periods. If you have replayed a game, click the Results button in the last of the string of replayed games. You can then use the View dropdown menu to easily switch between games.

Figure 1: Per-round Summary Tables

Each period, we present two tables (Figure 1). The first summarizes the game: the amount of money to be divided, the rate at which this amount shrinks per round, and the number of rounds. The second table summarizes outcomes: average payoff by player type, number of groups reaching an agreement, and the average round of an agreement (conditional on reaching an agreement).

Figure 2: Per-round Proposal Summary

In addition, we present a graph (Figure 2) summarizing proposals made in each period. With dollars on the vertical axis, we depict average proposal, with responder’s allocation stacked upon the amount the proposer proposes to keep. For each round, we present this information for both accepted and rejected proposals.

If your students played a multi-period game, we present a similar graph summarizing round 1 proposals. To access this graph, use the Go To dropdown menu to switch between periods.

If you have used the Replay, you can compare round 1 proposals across a string of replayed games. To access this graph, you must click the results button in the last of these replayed games. You then choose Multi-Game Summary from the View dropdown menu.

Equilibrium Prediction

In this section, we summarize equilibrium predictions. For full deriviations, see, for example, Tadelis (2013).2

In the finitely repeated game, the subgame-perfect Nash equilibrium is found by backward induction. We assume a player accepts a proposal if indifferent. In the final round, the responder accepts the proposal where proposer keeps everything. This highlights the last-mover advantage. In the preceding round, the other player (now the proposer) offers the recipient the payoff the recipient would receive in the final round, and keeps the remainder. This remainder (the difference between pie size in successive rounds) is the current-mover advantage. This proceeds backward until round 1.

We define the discount factor, \(\delta=(1- \text{Shrink Rate})\), and the total number of rounds, \(T\). The equilibrium amount that player 1 keeps in round 1 (\(\text{keep}_1^*\)) depends on whether the number of rounds is odd (she has the last-mover advantage) or even (her partner does). Under the assumption that amounts need not be integers, we have

$$\text{keep}_1^* = \left\{ \begin{array}{rl} \frac{1+\delta^T}{1+\delta} & \text{if number of rounds is odd;}\\ \frac{1-\delta^T}{1+\delta} & \text{if number of rounds is even}. \end{array} \right. $$

This approximates the equilibrium when values are constrained to whole dollars (as with MobLab), and highlights that the difference in payoffs decreases that as the number of rounds increases (i.e., increasing the number of rounds decreases the last-mover advantage). The effect of changes to \(\delta\) is more nuanced. Decreasing \(\delta\) increases the first-mover advantage, decreases the second-mover advantage, and thus, may increase or decrease player 1 payoffs when the number of rounds is odd (and player 2 has the second-mover advantage).

The indefinitely repeated game (Rubinstein, 1982)3gets rid of the second-mover advantage. With Minimum # of Rounds=1 and Ending Probability (%)<100, the discount factor is \(\delta=(1-\text{Shrink Rate})(1-\text{Ending Probability})\). In the unique subgame-perfect equilibrium \(\text{keep}_1^*=\frac{1}{1+\delta}\). With only an effect on the first-mover advantage, decreasing \(\delta\)—either by increasing the Shrink Rate, or increasing the per-round Ending Probability equilibrium payoff inequality increases.

1. 73 in the case that players are assumed to reject proposals when indifferent.
2. Tadelis, Steven (2013). Game Theory: An Introduction, Princeton University Press.
3. Rubinstein, Ariel (1982). “Perfect Equilibrium in a Bargaining Model.” Econometrica, 50(1): 97–110.
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