Coordination game
In game theory , a game in which the actors can achieve the highest payouts by coordinating their behavior is called a coordination game . In contrast to many strategic situations, the focus of these games is not conflict, but cooperation. The introduction of the term coordination game is generally attributed to Nobel Prize winner Thomas Schelling , although games of this type have been described and studied before. The choice of uniform technology standards can be seen, for example, as an application of coordination games in practice.
General
The essence of a coordination game is that there are several strict Nash equilibria and no other non-strict equilibria in pure strategies . Since none of these equilibria can a priori be preferred over the other, the need for coordination arises. The strategies form strategic complements. To illustrate this type of game, you usually find a bimatrix , which consists of two players with two strategies each. The principle of coordination games can, however, be extended to more than two players and / or more than two strategies.
| Left | Right | |
|---|---|---|
| Above | A, a | B, b |
| Below | C, c | D, d |
A coordination game arises here if A> C and D> B for player 1 (row player) and a> b and d> c for player 2 (column player). It follows that (top, left) and (bottom, right) form the two Nash equilibria in pure strategies. In addition, there is a mixed Nash equilibrium in coordination games , which results from a mixture of the two strategies of each player.
Examples
In most of the examples, the equilibrium arises from the players playing "the same" strategy. However, this is not absolutely necessary. Coordination games rely on players trying to coordinate their behavior, not on trying to "do the same". If, for example, different countries specialize in different branches of industry in order to realize increasing economies of scale , these countries try to prevent them from doing exactly the same thing through coordination. In addition, the naming of strategies or the swapping of strategy indices has no influence on the outcome or the classification of a game.
Typical coordination game
One of the first examples of coordination games comes from Thomas Schelling. Two people lose themselves in a crowd without having previously agreed on a meeting point for this case. Assuming there are two possible places (A and B) where they can meet again, both of them now have an interest in going to the same place.
| Place A | Place B | |
|---|---|---|
| Place A | 1, 1 | 0, 0 |
| Place B | 0, 0 | 1, 1 |
Coordination game with a conflict of interest
A situation where the players disagree on which point of balance to coordinate on is called a conflict of interest coordination game. The best-known example of such a situation is the battle of the sexes .
| Soccer | theatre | |
|---|---|---|
| Soccer | 2, 1 | 0, 0 |
| theatre | 0, 0 | 1, 2 |
In this game, a man (row player) and a woman (column player) want to spend the evening together. Without the opportunity to vote before the evening, they have to decide independently whether to go to the theater or to the football stadium. The man prefers the soccer game, the woman the theater, but the main thing is that they do something together and not separately.
Coordination game with common interests
If there is no conflict of interest in the above situation, but both prefer to watch the soccer game, for example, the result is a game with common interests. The name can be traced back to Thomas Schelling, who called them pure common-interest games .
| Soccer | theatre | |
|---|---|---|
| Soccer | 3, 3 | 0, 0 |
| theatre | 0, 0 | 1, 1 |
In a coordination game with equal interests, all equilibria are always Pareto-dominated by another equilibrium . In practice, however, it can happen that a Pareto-dominated equilibrium is established. In this case, one speaks of coordination errors or coordination failure, although a Nash equilibrium has been realized.
(Theater, theater) and (football, soccer) are the two Nash equilibria in pure strategies in this game and therefore equilibrium points of equilibrium in game theory. However, balance (soccer, football) is preferred by both players. So you can gain at the same time against the Pareto-dominated equilibrium (theater, theater).
Under the name win-win games this version of coordination games is often played on Manager seminars to show how to achieve better results through cooperation.
Risk in coordination games
A conflict between security and cooperation in coordination games usually arises when the players have a strategy that guarantees them a safe payout outside of the payout-dominant equilibrium, while the payout-dominant equilibrium strategy gives them a lower one in the event that coordination does not take place Brings in payout. The deer hunting game represents exactly such a situation. In this game the players can hunt a deer together or each of them can hunt a hare on their own. However, a hunted stag is preferred.
| deer | Hare | |
|---|---|---|
| deer | 4, 4 | 0, 3 |
| Hare | 3, 0 | 3, 3 |
Balance choice
The concept of the Nash equilibrium shows that there are combinations of strategies that are not worth deviating from. However, it does not say to which point of equilibrium the players are coordinating, or whether an equilibrium is realized at all. This section therefore deals with various concepts that try to enable coordination. Some of these are real decision-making situations that are not always to be viewed as normative solution concepts .
Focal points
The term focal point can be traced back to Thomas Schelling. It is assumed that a strategy or a particular balance combination stands out and is achieved through the anticipation of those involved. However, this is in part more based on imagination, fantasy, analogies, or aesthetics than logic . May just as well as conventions , different cultures or whether the players know each play a role. Game theory tries, however, to develop solutions and equilibrium concepts that lie within the game and are not given outside of the game and for which no prior knowledge of the other players is necessary. Therefore, the principle of focal points can only be viewed to a limited extent as a game-theoretical solution concept. However, experimental results in which participants were asked to give the same answers indicate the existence of focal points. Thomas Schelling himself provides some results on this. For example, 40% of respondents gave the number 1 when asked about a positive number. When choosing between “heads” or “tails”, 86% opted for “heads”.
Risk dominance
The concept of risk dominance , which was introduced by John C. Harsanyi and Reinhard Selten , is based on the intuitive perception that certain balance strategies are riskier than others, taking into account the uncertainty of a game. What is essential here is the rationality of the players and the probability with which they believe that another player will play a certain strategy. An extreme example of this is the deer hunting game. Since the equilibrium (stag, stag) dominates the equilibrium (rabbit, hare) Pareto, it should be assumed that rational players always prefer this equilibrium. The problem with this, however, is that if a player hunts a rabbit, they will definitely get a payout of 3, but if they choose to hunt a deer they run the risk of missing out. The payout that he receives when he goes hunting a deer depends on what the other is doing, or, to put it another way, how likely he is to believe that the other would also like to kill a deer. For this specific example this means that if for any reason one player believes that the other is going to hunt deer with a probability of less than 0.75, he will prefer to hunt a hare alone, as this will bring him a higher expected payout . In this case, a coordination error would result from pessimistic assumptions. The probability assignment shows that the concept is closely linked to that of the best answer , as the players try to find the best answer to every behavior of the opponent in order to minimize their risk.
Communication before the game
At first glance, the problem of being able to achieve a ( Pareto-optimal ) equilibrium in coordination games seems to be based on the fact that the players cannot communicate with one another. In game theory, however, communication can only be represented in a stylized way, which makes a modeled form of communication necessary. The simplest form of modeling is to give a player the ability to send a message or a signal . This message can consist, for example, of one player telling the other which strategy he will choose in the upcoming game. However, since such declarations of intent are not binding, in game theory one speaks of cheap talk in such cases, i.e. any statements that neither cause direct costs nor are verifiable. However, this leads to two new problems: Since the declaration of intent is not binding and the recipient does not know whether he can trust the sender, a new coordination game arises. And if the sender keeps his promise and the recipient believes him, this results in a situation in which the sender can choose a balance.
When modeling communication, however, both players can also be given the opportunity to send signals. One can assume that if the signals are sent at the same time, the players will also play this equilibrium by choosing the same signal in the next game. In the event that different signals are sent, the players then act as if this form of "communication" never existed. A study examining coordination games with a Pareto-dominant balance came to the conclusion that communication before the game is quite capable of solving a coordination problem. With two-way communication, the Pareto-dominant balance was achieved in over 90% of the cases, while one-way communication only had a success rate of 53% with regard to the Pareto-dominant balance.
Outside option and forward induction
Giving a player the ability to receive a safe payout instead of playing the coordination game is known as the outside option. This results in a two-stage game that can be solved using forward induction. If the outside option is high enough - it has to dominate a strategy of the coordination game - it has a credible signaling effect. If accepted, the game can be solved because rational players never play dominated strategies. That way, if the players anticipate this, they will always be able to strike a balance. Applied to the deer hunt example, player 1 could be given an outside option with a payout of 3.5. This now dominates the hare hunting strategy. This means that if he enters the game and rejects the outside option, he will not play the rabbit hunt strategy. Player 2 will anticipate this and also play the deer hunting strategy. In this example, coordination to the Pareto-dominant balance (deer hunting, deer hunting) is possible through forward induction.
Network effects
There are goods whose usefulness increases the more people own and use them. This means that the benefit depends on the number of consumers of a good. This effect is known as the network effect. This effect is often used in technical products. The greater the number of people who also own a phone, the more useful a phone is to the owner. The need for coordination in technical products arises when there are several competing systems. So decide compatibility and the ability to exchange data over the decision for which hardware , for example, chooses the computer purchase. The more frequently a certain type of hardware is in circulation, the more software there will be for it and vice versa. Therefore, the actors strive to coordinate on the same hardware-software combinations, as this increases their usefulness within the network. This can lead to a so-called lock-in situation , that is, if you commit yourself to the most widespread system, this state will “lock in” because nobody has any incentive to deviate from it. Due to the lock-in situation, "worse" systems can also prevail, as technically superior products no longer have a chance of spreading. Another example of network effects are the VHS and Beta video systems , of which VHS eventually prevailed. The focus here was on the possible exchange of video cassettes. Video stores and video users increased here their benefits through the coordination to a system.
It is of course also possible that the more people decide the same, the less the benefit. In this case one speaks of negative network effects. An example of this is the overloading of telephone networks when too many people within a telephone network try to call at the same time. Crowding games are precisely those games in which the benefit decreases the more players choose the same strategy. An example of a crowding game is road traffic. On the way from one city to another one can either drive on the motorway or on country roads. The route can be covered faster on the freeway than on the country road, but the more cars drive on the freeway, that is, the more players choose this strategy, the longer the journey time. So it can happen that you get to your destination faster on the country road. Another example of negative network effects are congestion games .
Discoordination games
Discoordination games are games in which one player tries to coordinate himself to a certain behavior, but the other player tries to avoid coordination.
Discoordination games have no equilibrium in pure strategy, as any player can increase his payout by deviating. The only equilibrium that exists in these games is created by randomizing the strategies, that is, playing a mixed strategy.
An example of this is the game Matching Pennies , in which two players independently choose heads or tails of a coin and place them on the table in front of them. If both have chosen the same thing, player 1 wins, otherwise player 2 wins.
| head | number | |
|---|---|---|
| head | 1, −1 | −1, 1 |
| number | −1, 1 | 1, −1 |
See also
literature
- Russell W. Cooper: Coordination Games. Complementarities and Macroeconomics . Cambridge University Press, Cambridge 1999, ISBN 0-521-57896-5 .
- Russell W. Cooper, Douglas V. DeJong, Robert Forsythe, Thomas W. Ross: Communication in Coordination Games . In: The Quartely Journal of Economics . Vol. 107, No. 2 , 1992, p. 739-771 .
- John C. Harsanyi, Reinhard Selten: A General Theory of Equilibrium Selection in Games . The MIT Press, Cambridge, Massachusetts 1988, ISBN 0-262-08173-3 .
- Michael L. Katz, Carl Shapiro: Network Externalities, Competition, and Compatibility . In: The American Economic Review . Vol. 75, No. 3 , 1985, pp. 424-440 .
- Judith Mehta, Chris Starker, Robert Sugden: The Nature of Salience: An Experimental Investigation of Pure Coordination Games . In: The American Economic Review . Vol. 84, No. 3 , 1994, p. 658-673 .
- Christian Rieck: Markets, prices and coordination games. Theoretical and experimental studies on the relationship between price and value . Physica-Verlag, Heidelberg 1998, ISBN 3-7908-1066-5 .
- Christian Rieck: Game Theory. An introduction . 8th edition. Christian Rieck Verlag, Eschborn 2008, ISBN 3-924043-91-4 .
- Thomas Schelling: The Strategy of Conflict . Harvard University Press, Cambridge, Massachusetts 1960, ISBN 0-674-84031-3 .
Web links
- Encyclopedia for Game Theory - Project of the University of Munich on the subject of game theory
Individual evidence
- ↑ Hal R. Varian: Fundamentals of Microeconomics . 8th edition. Oldenburg Verlag, Munich 2011, ISBN 978-3-486-70453-2 , p. 601 .
- ↑ a b c d e f g Christian Rieck: Markets, prices and coordination games, theoretical and experimental studies on the relationship between price and value . Physica-Verlag, Heidelberg 1998, ISBN 3-7908-1066-5 , p. 89-112 .
- ↑ a b c d e f g h Russell W. Cooper: Coordination Games, Complementarities and Macroeconomics . Cambridge University Press, Cambridge 1999, ISBN 0-521-57896-5 , pp. 1-18 .
- ↑ a b c d Christian Rieck: Game theory, an introduction . 8th edition. Christian Rieck Verlag, Eschborn 2008, ISBN 3-924043-91-4 , p. 58-80 .
- ↑ a b c d Thomas Schelling: The Strategy of Conflict . Harvard University Press, Cambridge, Massachusetts 1960, ISBN 0-674-84031-3 , pp. 54-58 .
- ^ Thomas Schelling: The Strategy of Conflict . Harvard University Press, Cambridge, Massachusetts 1960, ISBN 0-674-84031-3 , pp. 291 .
- ^ Judith Mehta, Chris Starker, Robert Sugden: The Nature of Salience: An Experimental Investigation of Pure Coordination Games . In: The American Economic Review . Vol 84, No. 3 , 1994, p. 658-673, here: pp. 666-670 .
- ^ A b John C. Harsanyi, Reinhard Selten: A General Theory of Equilibrium Selection in Games . The MIT Press, Cambridge, Massachusetts 1988, ISBN 0-262-08173-3 , pp. 82-90 .
- ^ Russell W. Cooper, Douglas V. DeJong, Robert Forsythe, Thomas W. Ross: Communication in Coordination Games . In: The Quartely Journal of Economics . Vol. 107, No. 2 , 1992, p. 739-771, here: pp. 748-756 .
- ↑ Michael L. Katz, Carl Shapiro: Network Externalities, Competition, and Compatibility . In: The American Economic Review . Vol. 75, No. 3 , 1985, pp. 424-440, here: pp. 424-425 .
- ↑ Igal Milchtaich: Generic Uniqueness of Equilibrium in Large crowding Games . In: Mathematics of Operations Research . Vol. 25, No. 3 , 2000, pp. 349-364, here: p. 349 .