   #copyright

Ultimatum game

2007 Schools Wikipedia Selection. Related subjects: Games

   Extensive Form Representation of a Two Proposal Ultimatum Game
   Enlarge
   Extensive Form Representation of a Two Proposal Ultimatum Game

   The ultimatum game is an experimental economics game in which two
   parties interact anonymously and only once, so reciprocation is not an
   issue. The first player proposes how to divide a sum of money with the
   second party. If the second player rejects this division, neither gets
   anything. If the second accepts, the first gets her demand and the
   second gets the rest.

Equilibrium analysis

   For illustration, we will suppose there is a smallest division of the
   good available (say 1 cent). Suppose that the total amount of money
   available is x.

   The first player chooses some amount in the interval [0,x]. The second
   player chooses some function f: [0, x] → {"accept", "reject"} (i.e. the
   second chooses which divisions to accept and which to reject). We will
   represent the strategy profile as (p, f), where p is the proposal and f
   is the function. If f(p) = "accept" the first receives p and the second
   x-p, otherwise both get zero. (p, f) is a Nash equilibrium of the
   Ultimatum game if f(p) = "accept" and there is no y > p such that f(y)
   = "accept" (i.e. p is the largest amount the second will accept). The
   first player would not want to unilaterally increase her demand since
   the second will reject any higher demand. The second would not want to
   reject the demand, since he would then get nothing.

   There is one other Nash equilibrium where p = x and f(y) = "reject" for
   all y>0 (i.e. the second rejects all demands that gives the first any
   amount at all). Here both players get nothing, but neither could get
   more by unilaterally changing their strategy.

   However, only one of these Nash equilibria satisfies a more restrictive
   equilibrium concept, subgame perfection. Suppose that the first demands
   a large amount that gives the second some (small) amount of money. By
   rejecting the demand, the second is choosing nothing rather than
   something. So, it would be better for the second to choose to accept
   any demand that gives him any amount whatsoever. If the first knows
   this, she will give the second the smallest (non-zero) amount possible.

Experimental results

   In many cultures, people offer "fair" (e.g., 50:50) splits, and offers
   of less than 20% are often rejected. These results (along with similar
   results in the Dictator Game) are taken to be evidence against the Homo
   economicus model of individual decisions. Since an individual who
   rejects a positive offer is choosing to get nothing rather than
   something, individuals must not be acting solely to maximize their
   economic gain. Several attempts to explain this behaviour are
   available. Some authors suggest that individuals are maximizing their
   expected utility, but money does not translate directly into expected
   utility. Perhaps individuals get some psychological benefit from
   engaging in punishment or receive some psychological harm from
   accepting a low offer.

   Based on fMRI studies of the brain during decision-making, different
   brain regions activate dependent upon whether the participating subject
   "accepts" or "declines" an offer. Since to "decline" means that neither
   receives any money, the responder is actually "punishing" the player
   who makes a low offer. Punishing activates the part of the brain that
   is associated with the dopamine pathway — i.e. it provides pleasure to
   punish. Hence, the subjects who refuse and punish in the process,
   possibly receive more pleasure from punishment than they would from
   accepting a low offer. This is, therefore, an expected utility argument
   where the currency is in pleasures received rather than goods or their
   associated values in money.

Explanations

   The classical explanation of the Ultimatum game as a well-formed
   experiment approximating general behaviour often leads to a conclusion
   that the Homo economicus model of economic self-interest is incomplete.
   However, several competing models suggest ways to bring the cultural
   preferences of the players within the optimized utility function of the
   players in such a way as to preserve the utility maximizing agent as a
   feature of microeconomics. For example, researchers have found that
   Mongolian proposers tend to offer even splits despite knowing that very
   unequal splits are almost always accepted. Similar results from other
   small-scale societies players have led some researchers to conclude
   that " reputation" is seen as more important than any economic reward.
   Another way of integrating the conclusion with utility maximization is
   some form of Inequity aversion model (preference for fairness).

   An explanation which was originally quite popular was the "learning"
   model, in which it was hypothesized that proposers’ offers would decay
   towards the sub game perfect NE (almost zero) as they mastered the
   strategy of the game. (This decay tends to be seen in other iterated
   games). However, this explanation ( bounded rationality) is less
   commonly offered now, in light of empirical evidence against it.

   It has been hypothesised (e.g. by James Surowiecki) that very unequal
   allocations are rejected only because the absolute amount of the offer
   is low. The concept here is that if the amount to be split were ten
   million dollars a 90:10 split would probably be accepted rather than
   spurning a million dollar offer. Essentially, this explanation says
   that the absolute amount of the endowment is not significant enough to
   produce strategically optimal behaviour. However, many experiments have
   been performed where the amount offered was substantial: studies by
   Cameron and Hoffman et al. have found that the higher the stakes are
   the closer offers approach an even split, even in a 100 USD game played
   in Indonesia, where average 1995 per-capita income was 670 USD.
   Rejections are reportedly independent of the stakes as this level, with
   30 USD offers being turned down in Indonesia, as in the United States,
   even though this equates to two week's wages in Indonesia.

Evolutionary game theory

   Other authors have used evolutionary game theory to explain behaviour
   in the Ultimatum Game. Simple evolutionary models, e.g. the replicator
   dynamics, cannot account for the evolution of fair proposals or for
   rejections. These authors have attempted to provide increasingly
   complex models to explain fair behaviour.

Sociological applications

   The split dollar game is important from a sociological perspective,
   because it illustrates the human willingness to accept injustice and
   social inequality.

   The extent to which people are willing to tolerate different
   distributions of the reward from " cooperative" ventures results in
   inequality that is, measurably, exponential across the strata of
   management within large corporations. See also: Inequity Aversion
   within companies.

   Some see the implications of the Ultimatum game as profoundly relevant
   to the relationship between society and the free market, with Prof.
   P.J. Hill, ( Wheaton College (Illinois)) saying:

          “I see the [ultimatum] game as simply providing counter evidence
          to the general presumption that participation in a market
          economy (capitalism) makes a person more selfish.”

History

   The first Ultimatum game was developed in 1982 as a stylized
   representation of negotiation, by Güth, Werner, Schmittberger, and
   Schwarze. It has since become the most popular of the standard
   Experiments in economics, and is said to be "catching up with the
   Prisoner's dilemma as a prime show-piece of apparently irrational
   behaviour."

Variants

   In the “Competitive Ultimatum game” there are many proposers and the
   responder can accept at most one of their offers: With more than three
   (naïve) proposers the responder is usually offered almost the entire
   endowment (which would be the Nash Equilibrium assuming no collusion
   among proposers).

   The “Ultimatum Game with tipping” – if a tip is allowed, from responder
   back to proposer the game includes a feature of the trust game, and
   splits tend to be (net) more equitable.

   The “Reverse Ultimatum game” gives more power to the responder by
   giving the proposer the right to offer as many divisions of the
   endowment as they like. Now the game only ends when the responder
   accepts an offer or abandons the game, and therefore the proposer tends
   to receive slightly less than half of the initial endowment.

   Retrieved from " http://en.wikipedia.org/wiki/Ultimatum_game"
   This reference article is mainly selected from the English Wikipedia
   with only minor checks and changes (see www.wikipedia.org for details
   of authors and sources) and is available under the GNU Free
   Documentation License. See also our Disclaimer.
