Subgame perfection in ultimatum bargaining trees
ABSTRACT In typical experiments on ultimatum bargaining, the game is described verbally and the majority of subjects deviate from subgame-perfect behavior. Proposers typically offer significantly more than the minimum possible and Responders reject “unfair” offers. In this work, we show that when the ultimatum bargaining game is presented as an abstract game tree, the vast majority of behavior is consistent with individualistic preferences and subgame-perfection. This finding raises doubts about theories that ignore the potential influence of social context and experiments that do not control for social context.
Subgame Perfection in Ultimatum Bargaining Trees1
Dale O. Stahl
Department of Economics
University of Texas at Austin
School of Management
University of Texas at Dallas
February 14, 2007
ABSTRACT: In typical experiments on ultimatum bargaining, the game is described verbally
and the majority of subjects deviate from subgame-perfect behavior. Proposers typically offer
significantly more than the minimum possible and responders reject “unfair” offers. In this
work, we show that when the ultimatum bargaining game is presented as an abstract game tree,
the vast majority of behavior is consistent with individualistic preferences and subgame-
perfection. This finding raises doubts about theories that ignore the potential influence of social
context and experiments that do not control for social context.
1 We are grateful to Gary Bolton, Colin Camerer, Jim Cox, David Cooper, Catherine Eckel, Dan Friedman, Al Roth,
Larry Samuelson, and Vernon Smith for helpful comments.
In ultimatum bargaining, one party (the Proposer) makes an offer to another party (the
Responder), who can accept or reject the offer. If accepted, the parties split a pie according to
the agreed terms; otherwise, neither party gets anything. The subgame perfect solution is for the
Proposer to offer the minimum feasible amount to the Responder, and for the Responder to
accept all positive offers. However, in laboratory experiments most Proposers offer close to an
equal split, and most Responders reject offers of less than 30%. See Guth, Schmittberger and
Schwarze (1982) for the original experiment, and Thaler (1988) and Roth (1995) for an overview
of ultimatum game experiments.2
There are several potential explanations for this deviation from subgame perfect
behavior. Subjects may have an intrinsic preference for an equal split (Bolton, 1991; Fehr and
Schmidt, 1999; Bolton and Ockenfels, 2000; Charness and Rabin, 2002, Stahl and Haruvy, 2002)
or they may exhibit envy (Kirchsteiger, 1994). Social preferences may be influenced by culture
(Roth et. al., 1991; Slonim and Roth, 1998), by the language in which the game is presented3, by
the seemingly arbitrary assignment into roles,4 or by mood (Capra, 2004). The desire to be
judged favorably can also trigger social norms. Even in double-blind experiments5, not all
subjects may be convinced that the experimenters cannot judge them, and even if convinced,
many subjects may believe in a divine being that is always observing and judging their behavior.
Subjects may be inexperienced at backward induction in the ultimatum game (e.g. Binmore, et.
al., 2002; Johnson, et. al., 2002) and may require learning to do better. Gale, Binmore and
Samuelson (1995) argue that learning is especially difficult in the ultimatum game and that
subgame perfection may be observed in the very long run (see also Roth and Erev, 1995). Jehiel
(2005) proposed an analogy-based approach in which first movers discretize their own action
space into analogy classes and offer the lower extreme point of the analogy class that would
result in acceptance. The reader, undoubtedly, can add to this list.
2 Experiments with sequential bargaining, such as Rapoport, Erev and Zwick, 1995 and Zwick and Chen, 1999, find
similar results—subjects behave in a manner consistent with a taste for fairness.
3 For example, in the Binmore, Shaked and Sutton (1985) paper, the subjects were told “You will be doing us a
favor if you simply set out to maximize your winnings.”
4 When adding a second stage (with subjects not being told until after the completion of the first stage), in which
players reversed roles, Binmore, Shaked and Sutton (1985) found that the second stage (which is an ultimatum
game) offers are in line with subgame perfection. Hoffman and Spitzer (1985) and Hoffman, McCabe, Shachat and
Smith (1994) found that subjects who were told they had earned (by winning a simple game) the right to be the
Proposer offered significantly smaller amounts.
5 Bolton and Zwick (1995) report that double-blind anonymity raises equilibrium play from 30% to 46%. They
conclude this effect is small and explains only 23% of non-equilibrium play.
In the current paper we show that a simple game-tree presentation of the ultimatum game
without suggestive language such as “dividing a pie” results in substantially more behavior
consistent with individualistic preferences and subgame perfection. There is other evidence that
extensive-form representation can result in systematically different behavior (e.g., Cooper and
Van Huyck, 2003; Deck, 2001; Schotter, Weigelt and Wilson, 1994).6
Related findings suggest that games described in terms of the complete set of contingent
payoffs to all players may result in less other-regarding behavior. For example, research by
Charness, Frechette and Kagel (2004) on the gift exchange shows drastically reduced reciprocal
behavior when payoff tables listing both players’ payoff for every wage-effort combination are
given in addition to the description of the game and the explanation of the game.
The present work is the first to demonstrate that such a presentation effect can result in
near subgame perfection in ultimatum bargaining. We describe and report on a series of
experiments designed to examine the effect of presenting an ultimatum games as a game tree
versus presenting it verbally. We also examine the behavioral effect of the potentially suggestive
Our results do not imply that the game-tree presentation is the only proper experimental
design for the ultimatum game. Rather, since presentation definitely affects behavior, the proper
design depends on the question being asked. If one is asking how well game theory predicts the
behavior in the abstract ultimatum game, then it is vital that the experimenter induce the payoff
structure of that game. Our results show that the game-tree presentation significantly reduces the
influence of unintended social context. On the other hand, if one is asking how people behave in
a socially rich context of dividing a pie that activates social norms and social judgments, then
obviously a context-sparse game-tree presentation would be inappropriate.
2. Experiments and Results.
Six experiments were conducted at the University of Texas in a Unix computer
laboratory. The subjects were third and fourth year undergraduate students and non-economics
graduate students with no previous experience with this game. The actual instructions for each
experiment are given in Appendices B-E.
The first experiment was a single-task design using the discrete ultimatum game tree
shown in Figure 1, which was presented as a hard copy handout. The payoffs points give the
percentage chance of winning $5. Three sessions were run with 14, 22 and 22 participants each.
6 Consistent with our hypothesis, Schotter, Weigelt and Wilson (1994) find that subjects are far less susceptible to
incredible threats in the extensive form relative to normal form representation of a particular 2x2 game.
Figure 1. The Discrete Ultimatum Game Tree
Each participant received $5 to compensate the participant just for showing up. The binary
lotteries provided an additional $0 or $5 for each participant. The second experiment replaced
the binary-lottery payoff with an “exchange rate” of 5 cents per point. Thus, the monetary values
of the potential offers were ($4, $1), ($3, $2), ($2, $3), and ($1, $4). Two sessions with
exchange-rate payoffs were run with 20 and 24 participants each.
Figure 2. Choices by Treatment Aggregated Across Sessions.
Responder Choice at A
Exp 1&2Exp 3&4
A ccept Reject
Proposer Choice Distribution
Exp 1&2 Exp 3&4
The choices by session are given in Appendix A. Figure 2 is a bar chart of the
distribution of Proposer choices and the distribution of Responder choices at node A (80:20)
aggregated across all sessions of experiments 1 & 2. This aggregation is justified by the fact that
the Fisher’s Exact Test for the difference between the proportion of proposer choices with
binary-lottery payoffs and with exchange-rate payoffs has a two-sided p-value of 1.00
(aggregating the B-D choices gives the same result). Similarly, for Responders, the largest
treatment difference is for the responder choice following the 20:80 offer, with a p-value of 0.43.
The p-value for the 80:20 offer is 1.00. Therefore, we cannot reject the hypothesis that the
binary-lottery, relative to exchange-rate payoff, makes no significant difference in Proposer or
Responder behavior. Summarizing experiments 1 and 2, 69% of Proposers chose the 80:20
branch, and only 4% of Responders rejected that offer. That 31% of Proposers made more
generous offers does not imply other-regarding preferences, since every Proposer choice can be
rationalized by some belief about the Responder. Hence, 96% of the observed behavior is
consistent with individualistic preferences.
This behavior is sharply different from the usual behavior in ultimatum games, but the
game is usually presented verbally without a tree7. To test whether the behavior in experiments
1 and 2 can be attributed to the tree presentation, we conducted two additional experiments using
a typical verbal description without a tree.
The third experiment maintained the singe-task feature, and the exchange-rate payoff
feature of the second experiment, but replaced the tree presentation with a typical verbal
presentation. The key phrase was: “The First Mover will choose a proposal on how to divide
100 points between him or herself and the Second Mover.” Three sessions were run with 18, 18
and 16 participants each. The fourth experiment was identical to the third except that instead of
exchange-rate payoffs, the payoffs were given directly in dollar terms: ($4, $1), ($3, $2), ($2,
$3), and ($1, $4).8 Two sessions were run with 24 participants each.
Figure 2 displays the distribution of Proposer choices and the distribution of Responder
choices at node A (80:20) aggregated across all sessions of experiments 3 & 4. This aggregation
is justified by the fact that Fisher’s Exact Test for the difference between the no-tree Proposer
choices with exchange-rate payoffs versus dollar payoffs has a p-value of 0.76. Further,
aggregating the B, C and D choices into one category, the Fisher’s Exact Test has a p-value of
7 When we refer to a pure tree format we allow for a verbal description of how to read tree payoffs but we do not
allow any bargaining-related or division-related terms.
8 Boles and Messick (1990, pp. 375-389) found that when responders physically saw dollar bills, low offers were
accepted more readily. As such, there is a possibility that presenting payoffs as dollars can have an effect.