Applied mathematics in Engineering, Management and Technology 3(1) 2015:158-165
www.amiemt-journal.com
The ultimatum game: A comprehensive literature review
Mohammad Mousazadeha
Milad Izadkhahb
a
School of Industrial Engineering, Faculty of Engineering, University of Tehran, Tehran, Iran
[email protected]
b
College of Economics, Management and Business, University of Tabriz, Tabriz, Iran
[email protected]
Abstract.
The ultimatum game (UG) is an outstanding game which shows how people behave
against rational strategies suggested by game-theory analysts. In the classic kind of
this game, one person plays as "proposer" who is supposed to split a determined
amount of money between himself/herself and another person who called
"responder". If the responder accepts the suggested amount of money, then the
proposal in implemented and the game is terminated but if the responder rejects the
proposal both will receive nothing and the game is terminated. Different extension
of this game i.e. dictator game, pirate game, three person game, etc. have been
introduced by researchers and also different factors such as size of stakes, fairness,
age, etc. have been studied in order to better analyze the behavior of the players
throughout the game. Since this game could be used as the base of many economic
and/or social theories and on the other hand a comprehensive review over extensions
and/or different factors have never been published, in this paper a comprehensive
review of the related papers covering around 30 years of time interval is provided.
Keywords: ultimatum game, dictator game, trust game, pirate game, fairness, equity.
1. Introduction
The classic UG has been first introduced by Güth et al. [1] in which the proposer has the authority to split a pie
between herself and a responder. Then the responder is free to accept or reject the proposal under this condition
thatif the responder accepts the proposal, the pie is divided according to the proposal; otherwise, both players
receive nothing. If one replaces $10 by the pie and also just integral multiple of $1 (i.e. $1, $2 to $10) be set as
the size of proposal, the sub-game perfect Nash equilibrium (or Stackelberg equilibrium) for this game will be a
game in which the proposer offers $1 to the responder and keep $9, and the second player accepts the proposal
since $1 is surely better than nothing.
The interesting aspect of this story comes from this fact that in the real situations, rarely the human behave as
discussed equilibrium. In fact, when experiments are conducted with human subjects, acceptance of small
amount of money is rarely observed [2]. This is an outstanding example of how human behavior differs from
game-theoretic analyses that use the “rational actor” model.
Copious evidence admitted that people are willing to “burn money” so that punish unfair behavior by others.
Since the full explanation of the sidelight reasons of this phenomenon is beyond the scope of this study, it is
preferred to just catch a quick glimpse about the main reasons. Among different explanations, some believe that
this event stem from inexperience of the subjects in this game, not having enough processing power to
thoroughly analyze the strategic interactions. They believe if people are allowed to play game many times and
accordingly develop expertise over time, they eventually learn and play the sub-game perfect Nash equilibrium.
However, different experiments have generally rejected this theory [3].On the other hand, some explanations
assert that cooperating of the subjects via some innate neurologically-based process make them receive positive
utility. Opposite to this reason, experiments have shown that people from different cultures play the UG
differently which support this fact that the internal process of decision making in the game is not innate at birth.
For instance, tribal members from the Machiguenga Indians in the Peruvian Amazon do not show any
inclinations towards the fairness while the sense of fairness is commonly observed in developed, industrial
societies [4].
This study consists of a systematic review of related empirical articles published between years 1982 to
2013.For this purpose, around 85 papers in this time interval were collected and among them around 40
empirical papers were approved qualified for being review. The review process includes two parts (1) to
classify different kinds of UG and find out the distinctions and (2) to identify the main explanatory variables
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which determine the outcome of the game. This will clarify convergence and divergence of the studies and will
broaden our knowledge of human behavior and its impact on economic theories and as a result to better channel
future research.
The rest of the paper is organized as follow. In section 2, different kinds of UG such as social ultimatum game,
dictator game, trust game, pirates game and three-person ultimatum game is described. Afterwards, in section 3
different factors and their impacts on UG are fully explained and finally, some concluding remarks are
presented in section 4.
2. Different kinds of ultimatum game
As first step to uncover literatures concerned with ultimatum game, we have to identify various kinds of this
game which are introduced by Scholars heretofore. We distinguished five different versions of ultimatum game
including: (1) social ultimatum game, (2) dictator game, (3) trust game, (4) pirates game and finally (5) threeperson ultimate game. In this section, we briefly review the characteristics of each abovementioned games and
also describe sub-perfect Nash equilibrium of this game (if any) and compare expected results with real results.
2.1. Social ultimatum game
Social ultimatum game (SUG) was first introduced by Chang et al. [2].The main difference of this game to the
classic game is that in SUG players can choose their partner among a society of agents, and engage in repeated
interactions of the classic game. As expected, human players do not behave according to simple equilibrium
analysis predictions. Almost no one attempts to offer only $1, and when it is attempted, the offer is summarily
rejected. Both the distribution of offer amounts and corresponding rejection rates generally resemble the
distributions observed in the standard ultimatum game. That is, we do not see a complete shift towards game
theoretically rational strategies: low offers are still rejected, and for the most part, near-fair offers are the most
common offers made. However, there are some clear differences. First, the mode of the offers in the SUG is $4
while the mode in the standard ultimatum game is $5. It shows that in the social ultimatum game players behave
more flexible.
Second, and perhaps more surprisingly, players actually offer more than $5 in a fair percentage of the time
(roughly 15%). This makes sense due to the repeated nature of the game. This kind of human strategizing is
exactly the creative and subjective modeling that humans are capable of performing, but that machines cannot
yet emulate. Humans intuitively understand that a high offer, say $8, makes a big impression on the recipient,
potentially causing the recipient to act more favorably in response. Interestingly, based on SUG, the web based
game which then is optimized for easy-to-use touch-based devices such as iPad is generated.
2.2. Dictator game
The dictator game (DG) is a simpler form of the ultimatum game, where one player (the allocator) receives an
amount of money and makes a unilateral decision how to divide it. The other player (the recipient) must accept
this split of the pie. Once again, on the assumption that people are only concerned with maximizing own
monetary payoffs, the unique Nash equilibrium is for the allocator to take all the money for himself/herself.
Two main conclusions emerge from the literature on this game (e.g. [5-8]). First, the equilibrium prediction is
not supported by most of the data, and second, results are very sensitive to the experimental procedure (see the
surveys in [9, 10]).
2.3. Trust game
The “trust game” (TG) or “investment game” (IG) was first proposed by Berger et al. [11]. In this game there
are two participants that are anonymously paired, both participant are given an amount of money, the first
player have to transfer some or all of his endowment to the second player(the amount sent may be zero), this
transfer is tripled by the experimenter and handed to the second mover, and finally the second player may return
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some or all of the received transfer. In fact, “first-mover transfers are interpreted as a manifestation of trust, and
second-mover transfers as a manifestation of trustworthiness [12]”.
Based on economic assumptions of rational self-interest, the amount of money transferred by first player would
be zero even if he or she is well informed about the design of the game [11], however actual results
acknowledge the deviation of real observed results from the predicted results by economic theories. The
findings of the real world experiments shows that amount of money sent back by first mover averaged slightly
over fifty percent of their original endowment [11].
2.4. Pirate game
Pirated game (PG) is considered as a multi-player form of ultimatum game, assumes players as self-interested
actors in human economic concepts. The game consists of five rational pirates namely A, B, C, D and E
between whom there is an order of superiority i.e. A is superior to B, who is superior to C, who is superior to D,
who is superior to E. In this game, the pirates are going to distribute 100 gold coins between each other and the
rule of distribution is as follow: the most senior pirate, suggest a method of allocation then the others including
the proposer vote on whether to accept this proposal or not. If they approved this distribution then the game is
finished and the coins are allocated based on the proposed method, otherwise the proposer will be killed
(eliminated from the game) and the next most senior pirate makes a new proposal to begin the system again.
Each player in this game have to take into account three different factors: first of all he or she must survives,
second factor is that each player should strive for maximizing the number of gold coins he/she receives and the
last is that each one prefers to eliminate the others if all other results would otherwise be equal [13].
2.5. Three-person ultimate game
There is also a wide recognition of bargaining situation with more than two players [14]. The three person
ultimatum game (3P-UG) as an extended of standard game has been investigated by scholars in which the
mutual behavior toward the proposer is affected by the presence of the third player. A three person ultimate
game, consist of a proposer and two responders, both responders simultaneously decide whether to accept or
reject the proposal. If both responders accept then all players' earnings are according to the proposal but even if
one of the responder (or both of them) rejects the proposal, all earns nothing.
In an experiment designed by Knez and Camerer [15], the responders' strategies were investigated when the
proposal made with asymmetric outside options, each responder is either not informed or informed about the
proposal in the parallel ultimatum game. The results of experiment indicated that there were kind of “betweenresponder payoff comparison” [14], means that each responder varies his/her minimal acceptable amount based
on the offer made to the other responder [15].
3. Factors studied
The reviewing of related articles, brought out a wide range of issues related to UG and its explanatory
factors.Weenumerate15 different variables claimed to influence the results of the game. The following section
presents these factors and also discusses their effects on the game.
3.1. Size of stakes
A reliable result in the literature was that the respondent's behavior is quite independent from size of the stake
([16-18]) but in the game designed by Anderson et al. [19],they find that sufficiently high stakes lead responder
behavior to converge almost perfectly to full acceptance of low offers, even in the absence of learning effect.
That study provides the first empirical support for the hypothesis of substantial stake effects in this game and
provides insights into what might happen in higher stakes games.
3.2. Aging
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It is also proven that age of the players can influence social-economic decision makings [20]. It is recognized
that older adults are less impulsive, sensation seeking and risk tolerant than younger adults. Older adults were
more likely to reject unfair divisions of money during an economic social-bargaining game and more likely to
make equitable divisions of money during social giving game [21].
3.3. Impact of groups
In many economic theories humans are assumed as a rational, self-interested decision makers, who attempt to
maximize its utility or economic profit [22]; many studies however, questioned these assumptions at least when
examining decisions made by individuals. There is much debate as to whether these economic human premises
are also problematic in a situation where individuals make a choice from the alternatives collectively which is
known as group decision making [23]; for example Kugler et al [24] by reviewing the literature of
collaborative decision making and comparing strategic behavior of groups and individuals in many games,
found that that group decisions are relatively closer to the game-theoretic assumption of than individual
decisions.
3.4. Information about features of the other player.
In a paper by Marchetti et al. [25], the impact of information about the proposer on the responder's decision is
studied. In their research, four levels of description (information) were transferred between proposer and
recipient which are (1) no information (control condition), (2) physical description including age, height, eye
color, hair color, dress style, (3) brief positive psychological description (generous condition) and finally (4)
brief negative psychological description (selfish condition).
The results show that exchange of information affects the game. The achieved results are shown in Figure 2. As
could be seen, the acceptance rate is significantly affected by both offer fairness as well as type of description
provided of the proposer. The results indicates that the information about features of the players may generate
the expectations about the outcome so it would causes violation of monetary self-interests [25].
100%
90%
90%
80%
76%
73%
70%
60%
53%
46%
50%
40%
40%
30%
23%
23%
20%
10%
0%
No descr.
Phys. descr.
Fair Offers
Gen. descr.
Self. descr.
Unfair Offers
Fig. 1. Acceptance rate for each ultimatum game group (Source: Marchetti et al. [25])
3.5. Anonymousness effect
Current body of the literature also emphasis on the impact of "anonymity" on ultimatum bargaining e.g. [7, 26].
Among the most recent related researches, Lamba and Mace [27] designed an experiment in which the
ultimatum game was played under three conditions as described below.
 A double-blind anonymous ultimatum game such that no one could link a decision to a participant, while
they were made explicitly aware of this set-up prior to making their decisions. All participants made their
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decisions in full privacy. This situation was called "Anonymous ultimatum game" in "anonymous context"
(Condition AA; Completely Anonymous),
 Similar to the first situation except that the fact that all participants were seated in the same room and made
their decisions independently, but in the presence and full view of all other participants. This situation was
called "Anonymous ultimatum game" in "public context" (Condition AP; Anonymous in Public),
 Similar to the second situation by this difference that all decisions were made public knowledge once all
participants had made their decisions, so that decisions could be linked to specific individuals. This situation is
called "Public ultimatum game" in "Public context" (Condition PP; Completely Public).
The results show that proposer offers did not vary with changes in context (i.e., there was no “eyes effect”) but
did vary with the degree of actual anonymity and the specific presence of known others. Responder behavior
did not vary with changes in context, degree of actual anonymity or the specific presence of known others [27].
3.6. Impact of delay and waiting time
In different studies, it was proofed that delay or waiting time does not have any effect on the game and only
amount of money plays an important role in acceptance or rejection of a proposal [28]. But in contrast, Grimm
and Mengel [29] expressed delaying acceptance decisions in the ultimatum game drastically increases
acceptance of low offers. In detail, in ordinary games at most 20% of low offers are accepted while by allowing
responder to respond 10 minutes later, the acceptance rate of low offers changes to 60–80%. The achieved
results are depicted in Fig 3.
Fig. 2. Acceptance rates with and without delay (Source: Grimm and Mengel [29])
3.7. Impact of culture
Culture is known as one the economic variables that influence economic performance at the national level.
Among the related papers, Chuahet al. [30] investigate the attitudes and experimental behavior of Malaysian
and UK subjects in order to shed light on the nature of culture and the mechanisms by which it affects economic
behavior. Also, Oosterbeek [31] find differences in behavior of responders (and not of proposers) across
different geographical regions.
3.8. Impact of fairness
There is sufficient evidence that fairness considerations affect economic behavior in many environments [32]. It
is investigated that in mutual gain bargaining situations anonymously players agree on rather equal outcomes.
Also, it is proved that identical offers in an ordinary ultimatum game will decrease rejection rates by responder
depending on the other available offers to the proposer [13]. It means that “a given offer with an unequal
distribution of material payoffs is much more likely to be rejected if the proposer could have proposed a more
equitable offer than if the proposer could have proposed only more unequal offers [32]”.
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3.9. Physical appearance
It was believed that physical appearance can influence the bargaining process in a number of situations. As an
example, one can refer to Solnick and Schweitzer [33] that conduct an experiment to test the effect of physical
attractiveness on ultimatum game decisions. Finding revealed no significant differences in the offers or
demands has been made by attractive and unattractive people. However the treatment toward attractive people
by others was different as they were offered more and more was demanded of them.
3.10. Geographical distribution
Reviewing the literature also reveals the increasing consideration of the impact of geographical differences on
the expected outcomes of ultimatum game. As an instance, Kohler [34] assessed the level of social cohesion in
some of the newly established communities in Zimbabwe by estimating average preferences for fairness in a
structural model of bounded rationality. For this purpose, the ultimatum game was played by 234 randomly
selected households in six traditional and 14 resettled villages almost two decades after resettlement. The
achieved results show that in two out of three distinct resettlement schemes studied, the resettled villagers
exhibit considerably higher degrees of fairness as well as rationality than those who live in traditional villages.
Also, these findings are conforming to the idea of a raised need for cooperation required in recommencement
[34].
In another research done by Boarini et al. [35], the experimental results of a “transcontinental ultimatum game”
implemented between India and France was presented. The bargaining took the form of standard ultimatum
games, but in one treatment Indian subjects made offers to French subjects and, in another treatment vice versa.
The findings show that French→Indian interactions usually ended up with unequal splits of money in favor of
French, while nearly equal splits were the most frequent outcome in Indian→French bargaining.
3.11. Players experienced war
According to the study by Bellows and Miguel [36] conducted in Sierra Leone, people who experienced more
violence during the civil war participate more in local collective action. These findings is also investigated and
reported in ultimatum game (see [37]). Also, it is proved that more exposure to war will result in more sharing
with neighbors in Burundi (see [38]) and more in-group inequality aversion in Georgia and Sierra Leone (see
[39]).
3.12. Kinship and family effect
Reciprocity and kinship are two main predictors of philanthropy. In order to assess the effect of individuals’
family and kin relations on philanthropic behavior in ultimatum game, Macfarlan [40] conduct a research in a
kin-based horticultural community in rural Dominica. Since different hypothesizes are examined in that
research and different results are achieved, here that results are not discussed in detail and the interested readers
are referred to the main paper.
3.13. Gender
Gender of both proposer and responder was predicted to affect the game. However, Kohler [41] does not find
any evidence for gender-related differences in the Zimbabwean villagers’ preferences. Although the reported
independency could be due to the small sample size, but that resettlement status affects the value villagers place
on equality significantly.
3.14. Effect of schizophrenia
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Schizophrenia is a severe mental disorder in which people interpret real situations abnormally and as a result
have problems in thinking and have poor emotional response. It was assumed that this disorder would impair
the patients' bargaining ability. Interestingly, Agayet al. [42] found that patient who suffer from general
schizophrenic did not completely exploit their strategic mental power as a "proposer" but their action as a
"responder" did not significantly different from ordinary people.
3.15. Effect of masculinity
It has been debated that testosterone and masculinity will have apparent effect in many human behaviors such
as financial risk preferences [43]. According to this discussion, Burnham [44] conducted a research in which the
impact of testosterone on ultimatum game rejections was explored and the results clarified that hightestosterone men reject low ultimatum game offers.
4. Conclusion
The ultimatum game is an exemplar of how human behave against rational strategies suggested by game-theory
and economic analysts. In the two-player UG (standard game), one plays as "proposer" and one plays as
"responder". The proposer decides to divide a given amount of money between both players. If the responder
accepts the proposal, then the proposal is implemented, otherwise both earn nothing. Although the expected
sub-perfect Nash equilibrium of the game is when the proposer suggests small proportion (near to zero) to the
responder and the responder accepts the proposal, in real situation this expectation do not happen and small
offers are rejected by responders.
This game have different extensions such as dictator game, social ultimatum game, three-person ultimatum
game, pirates game and trust game and on the other hand effect of different factors such as age, gender, culture,
delay in response etc. in the game has been widely studied by different scientist start from 1982 till now. Since
the observed results of this game in real situation could be utilized as the base for many economic and social
theories, and also, to the best of our knowledge there were no comprehensive review over this game, in this
review paper we provide the readers with concise review of the discussed game, its main extensions and studied
factors from 40 selected papers published between 1982 to 2013 covering around 30 years time interval. This
review could be a starting point for the interested readers who want to study different aspects of the ultimatum
game in detail.
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