Ultimatum game

The ultimatum game (also called ultimate negotiation ) is one of the practical applications of game theory for economic and behavioral research . It was implemented experimentally by Werner Güth et al. (1982). The ultimatum game is often used as a laboratory experiment to research altruism or egoism . In different variations of the game it is investigated to what extent the person maximizes the benefits resulting from the object of the game and to what extent the person also includes other interests in his decisions. Examples of other interests to be taken into account are the maintenance of the rules of the game that are useful to him or the community, and cultural customs such as the sense of justice, as well as the effect of one's own personality image on fellow players and observers. The ultimatum game is also successfully used for neurobiological experiments - among others - to examine the effects of (damaged) brain areas on behavior, for example. A multiplayer version is the so-called Pirate game .

Basic form of the ultimatum game

An actor A ₁ is a good set (eg. As money) available. Of this he has some select ( ) and another actor A ₂ offer. If he rejects the part offered to him, A _{1 also has} to forego his part and both go away empty-handed. If A ₂ accepts, he receives the offer and A ₁ receives the rest . ${\displaystyle c}$ ${\displaystyle s}$ ${\displaystyle 0\leq s\leq c}$ ${\displaystyle s}$ ${\displaystyle c-s}$

One objective of the game is for player A ₁ to maximize his winnings in the form of money. (He could, however, also have other goals, for example to distribute the money “fairly”. In the present discussion, profit maximization is assumed.) Player A _1's goal is not necessarily known to player A ₂ . But he can suspect it on the basis of social experience. In the standard version of the ultimatum game, the players are not known and cannot communicate with each other. The two players have no consequences to fear, apart from a failure to win.

The game-theoretic solution for profit-oriented rational players is that A ₁ offers only the smallest possible part of the sum (e.g. "1 cent") because he knows that a rational player A ₂ in the sense of individual utility maximization will offer this small amount a payoff of zero prefer and will therefore agree (with a payoff of zero would result for a ₂ no advantage, so he could reject). A ₁ has thus minimized its investment and maximized the share to be paid out to itself. In experiments, however, many players A ₂ did not behave rationally in this sense, but rather refused a small win than accept an unfair division. Offers below around 15% of the total are usually rejected, so that the provider also goes empty-handed. The division is slightly different. On average, A ₁ leaves 30% of the goods to A ₂ . A division that differs drastically from the “rational” division is practically always common. ${\displaystyle c}$ ${\displaystyle s>0}$

If A ₁ already owns the goods and can give part of it to A ₂ , whereby A ₁ receives additional profit as before , there are three interesting solutions: ${\displaystyle c}$ ${\displaystyle s}$ ${\displaystyle c-s}$

1. A ₁ gives so that A ₁ loses nothing and A ₂ wins without penalizing A ₁ , half of the good of A ₁ . ${\displaystyle s={\tfrac {c}{2}}}$
2. A ₁ is such that both A ₁ and A ₂ , win. ${\displaystyle s={\tfrac {c}{3}}}$ ${\displaystyle {\tfrac {c}{3}}}$
3. If A ₂ already owns the good , the relative profit of both is the same with the acceptance of the offer with the profit factor . ${\displaystyle d}$ ${\displaystyle s=c\cdot {\tfrac {d}{c+2d}}}$ ${\displaystyle 1+{\tfrac {c}{c+2d}}}$

If the good of A _{1 is} greater than that of A ₂ , the third solution, based on the absolute profit, is disadvantageous for A ₂ . Only when the good of A _{2 is} greater than that of A ₁ , solution 3 is advantageous for A ₂ - and vice versa: the person who has more good will try to implement solution 3, the other will strive for solution 2. If the good of A ₁ and A _{2 is the} same, then the solutions 2 and 3 are identical.

If your own good is smaller than that of your playing partner, it is advantageous to hide your own good from your playing partner as much as possible in order to increase the probability of implementing solution 2. In contrast, it is important for the richer playing partner to determine the entire property of the poorer playing partner in order not to arouse suspicion of unfairness when attempting to enforce solution 3. The yardstick for fairness is culture-dependent.

If a division takes place among several, then the willingness to accept smaller sums increases.

The division also partly depends on the specific cultural customs.

In economics , a Pareto-optimal equilibrium describes a distribution of scarce goods in which no one involved can be better off without making another worse off.

Variant: dictator game

A dictator game is a variant of this game in which A ₂ cannot refuse the offer. With this variant, one would only expect the smallest possible offer if minimizing losses in the domain of the obvious utility function (giving away as little money as possible) were the only relevant behavior. If a minimization of losses (“avarice”) is not predominantly observable in this area, it must be examined which other utility functions (e.g. maintaining the willingness to cooperate for future games) influence the decisions of the participants in this game.

Maximizing benefits on different levels

If the behavior that does not follow individual rationality leads to a maximization of the benefits (e.g. money, resources) for a group, then collective rationality is played. If no benefit maximization (or at least loss minimization) can be observed in a game, then there is either no rationality whatsoever, or the game was played for a benefit that has not yet been recognized, for example in a meta game about rules of the game that secure future benefits. Ultimatum games are well suited to illustrate this fact and to demonstrate deparadoxification in game analysis.

In ultimatum games, the difference between individual and often repeated games becomes clear. Here the individual game is played in communities in such a way that the overall game consisting of individual games maximizes the benefit of the community or minimizes losses. This can result in an efficient sharing and distribution of resources within this community.

Empirical results

Developed country players - mostly undergraduate students from the United States, Europe, and Asia - typically offer between 40% and 50% of the amount to the second player, and offers below 30% are typically turned down by the second player. Martin A. Nowak et al. (2000) were able to predict these results in a model if this takes into account reputation , i.e. provides information about the behavior of a player in the past. Nowak et al. Concluded that the ultimatum game showed a universal human tendency towards fair and punitive behavior. Since the experiments are played without reputation, but the results only resemble those of simulations with reputation, Nowak assumed, among other things, that fairness and punishment emerged in an evolutionary context in which interactions without reputation were not relevant to fitness .

In other cultural contexts, however, the ultimatum game had different results. Joseph Henrich et al. Conducted the game with several randomly selected small ethnic groups. It was shown that the players from the industrialized countries represented the high extreme of the range of offers and rejections. In the smallest companies very low offers were made and they were not rejected. These results were thus similar to the simulations by Nowak et al before they considered reputation. In the USA, the dictator game offered on average almost twice (> 45%) as many offers as the Hadza (<30%). The ultimatum game produced similar results, and the rejection threshold in the USA is higher than in any other society examined. Analysis of this data shows that the degree of market integration (proportion of the food consumed that was bought) and the degree of religiosity both independently predict higher offers. In other words, the results of the ultimatum game in industrialized countries are not human universals, but specifically the cultural development of these societies. Henrich et al. Assume that complex free-market societies are not possible without a high degree of cooperation with strangers.

literature

• Stephen D. Levitt, Steven J. Dubner: Superfreakonomics. Riemann, 2010, ISBN 978-3-570-50122-1 .
• Karl Siegmund, Ernst Fehr , Martin A. Nowak : Sharing and helping / origins of social behavior. In: Spectrum of Science. Dossier. Issue 5, 2006, p. 55. ( PDF version of the article from Spectrum of Science, Issue 3/2002 )
• Werner Güth, Rolf Schmittberger, Bernd Schwarze: An experimental analysis of ultimatum bargaining. In: Journal of Economic Behavior & Organization. Volume 3, Issue 4, December 1982, pp. 367-388.

Individual evidence

1. ↑ Gerd Böhmer: Neuroeconomics (Neuroeconomics): Neural mechanisms of economic decisions . Johannes Gutenberg University , Mainz 2010. ( Abstract with download link, PDF, 10.8 MB ( Memento of the original dated February 23, 2014 in the Internet Archive ) Info: The archive link was inserted automatically and has not yet been checked. Please check the original and archive link according to the instructions and then remove this note. ) 
2. ↑ MA Nowak, KM Page, K. Sigmund: Fairness versus reason in the ultimatum game. (PDF; 92 kB). In: Science. Volume 289, 2000, pp. 1773-1775.
3. ^ ^{A b} J. Henrich, S. Heine, A. Norenzayan: The Weirdest People in the World? In: Behavioral and Brain Sciences. (PDF; 1.2 MB). Volume 33, 2010, pp. 61-135.
4. ↑ J. Henrich, J. Ensminger, R. McElreath, A. Barr, C. Barrett, A. Bolyanatz, JC Cardenas, M. Gurven, E. Gwako, N. Henrich, C. Lesorogol, F. Marlowe, DP Tracer , J. Ziker: Market, religion, community size and the evolution of fairness and punishment. (PDF; 211 kB). In: Science. Volume 327, 2010, pp. 1480-1484.