Strategy (game theory)

Under a strategy of a player is meant in game theory a complete plan on how the players will behave in every possible game situation. The strategy fully describes the behavior of a player in the game.

Examples

In the game of " rock, scissors, paper ", a player's behavior can be fully described by indicating which symbol he chooses. Each player has exactly the three possible strategies “scissors”, “stone” and “paper”.
In order to specify the strategy of a player in the game of chess , one would have to determine how the player would behave in each move in response to all possible courses of the game up to that point. A strategy of the white player would therefore be indicated in the following way: “Play x1 first; if you have drawn x1 and then black draws y11, draw x11 on the second move; if you've drawn x1 and black draws y12, draw x12; if etc .; ...; if you had drawn x2 and black y21, draw x21; ...; if you have drawn x1, black has drawn y11, you then move x11 and black then y111, then move x111 ”etc. You can see that this is practically impossible (except for“ trivial ”strategies). For the theoretical analysis of the game of chess (e.g. to prove that there must be a strategy for one of the two players with which he does not lose), this conceptual construction makes sense.

Set of strategies, course of the game, normal form

Since the strategy fully describes the game behavior of a player, the course of the game and thus the payouts of the individual players are determined (in a game without random external factors, e.g. without rolling the dice) if you know which player is playing which strategy. (In the example above: if we know that player A is playing “stone” and player B is playing “paper”, we know that B will win; if they play for one euro, A will lose one euro and B a payout of one Formally: The combination of strategies ("stone", "paper") leads to the payout vector (-1, 1).)

The set of all strategies of a player is called the set of strategies (often abbreviated with , where denotes the player). In the above example “Scissors, Rock, Paper” the strategy sets of all players are the same, namely ${\ displaystyle \ Sigma _ {i}}$ ${\ displaystyle i}$

{\ displaystyle \ Sigma _ {i} = \ {{\ mbox {scissors}}, {\ mbox {stone}}, {\ mbox {paper}} \}.}

A game like “scissors, rock, paper”, in which all players draw once and at the same time, can be formally described by specifying the amount of strategies for the individual players and the payoff function that determines the payouts for each strategy combination. If a game is defined in this way, one speaks of a game in normal form .

If the players do not draw at the same time (such as when playing chess), such a simple description is often not sufficient; one then has to resort to the extensive form . Since all possible reactions of the other players have to be taken into account, the strategies in such games can be very complicated.

Pure and mixed strategies

Strictly speaking, so far only pure strategies have been mentioned, i.e. H. of strategies in which each player clearly decides on a certain action. However, games in pure strategies often have no equilibrium. "Scissors, rock, paper", for example, has no (Nash) equilibrium in pure strategies: If one player clearly decided on a symbol (such as "paper"), the other player would choose the better one (here "scissors"), what the first one anticipates and therefore will not commit itself.

A way out is offered by mixed strategies in which the player does not commit to a pure strategy, but mixes several pure strategies according to a probability distribution . Mixed strategies in the game “Scissors, Stone, Paper” would be (among many others, of course) for example “choose 'stone' and 'scissors' with a probability of 1/2” or “choose 'scissors', 'stone' and 'paper' respectively with probability 1/3 ". If you play “scissors, rock, paper” for a fixed amount of money and the players want to maximize their expected payout , a balance results from both players playing this “third strategy”. As soon as one of the players plays the strategy of thirds, it does not matter which strategy the other player chooses for the expected payout. In contrast, with any other strategy, the opponent can choose a strategy that delivers an expected value that is more favorable for him than the strategy of thirds. Conversely, this means for the player that deviating from the strategy of thirds means a disadvantage for him if it becomes known to the opponent.

Continuous strategy

If the (infinite) number of actions (and thus the strategies) of a player in a game cannot be counted , one speaks of continuous strategies. An example could be a game in which two players must choose a number from the real numbers between 0 and 1, with the one with the larger number winning. (To rule out the obvious choice 1 here, let 1 be forbidden in the game.)

In games with continuous strategies, the game is often characterized by so-called reaction functions. The Nash equilibrium ( i.e. the tuple that consists of the best answers of all players) is determined from the points of intersection of the reaction functions of the players.

Strategies of nature

Games with non-deterministic elements, so-called games with random moves (such as dice games), can be understood as strategic games without random moves, in which chance ( nature ) takes part and in which chance itself plays a mixed strategy (a dice would therefore be the strategy "Choose every number with a probability of 1/6"). The "real" players anticipate this strategy of nature in their decisions. In contrast to a “real” player, it cannot be assumed that nature is “strategically”, i. H. behaves rationally .

literature

Elwyn R. Berlekamp, Winning ways for your mathematical plays , 2nd edition (Natick, Mass .: AK Peters, 2003)
Andreas Dorschel , 'Is strategic action in need of addition?', Archives européenes de sociologie XXX (1989), pp. 123–149
John A. Sokolowski et al. Catherine M. Banks (Ed.), Modeling and simulation fundamentals: theoretical underpinnings and practical domains (Hoboken, NJ: Wiley, 2010)
Martin J. Osborne, An introduction to game theory (New York - Oxford: Oxford University Press, 2009)