Outcome (game theory)

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In game theory, the outcome of a game is the ultimate result of a strategic interaction with one or more people, dependant on the choices made by all participants in a certain exchange. It represents the final payoff resulting from a set of actions that individuals can take within the context of the game. Outcomes are pivotal in determining the payoffs and expected utility for parties involved. [1] Game theorists commonly study how the outcome of a game is determined and what factors affect it.

Contents

In game theory, a strategy is a set of actions that a player can take in response to the actions of others. Each player’s strategy is based on their expectation of what the other players are likely to do, often explained in terms of probability. [2] Outcomes are dependent on the combination of strategies chosen by involved players and can be represented in a number of ways; one common way is a payoff matrix showing the individual payoffs for each players with a combination of strategies, as seen in the payoff matrix example below. Outcomes can be expressed in terms of monetary value or utility to a specific person. Additionally, a game tree can be used to deduce the actions leading to an outcome by displaying possible sequences of actions and the outcomes associated. [3]

Payoff Matrix Example

Strategies of Player A

Strategies of Player B
12
1A1, B1A1, B2
2A2, B1A2, B2

A commonly used theorem in relation to outcomes is the Nash equilibrium . This theorem is a combination of strategies in which no player can improve their payoff or outcome by changing their strategy, given the strategies of the other players. In other words, a Nash equilibrium is a set of strategies in which each player is doing the best possible, assuming what the others are doing to receive the most optimal outcome for themselves. [4] It is important to note that not all games have a unique nash equilibrium and if they do, it may not be the most desirable outcome. [5] Additionally, the desired outcomes is greatly affected by individuals chosen strategies, and their beliefs on what they believe other players will do under the assumption that players will make the most rational decision for themselves. [6] A common example of the nash equilibrium and undesirable outcomes is the Prisoner’s Dilemma game. [7]

Choosing among outcomes

Many different concepts exist to express how players might interact. An optimal interaction may be one in which no player's payoff can be made greater, without making any other player's payoff lesser. Such a payoff is described as Pareto efficient, and the set of such payoffs is called the Pareto frontier.

Many economists study the ways in which payoffs are in some sort of economic equilibrium. One example of such an equilibrium is the Nash equilibrium, where each player plays a strategy such that their payoff is maximized given the strategy of the other players.

Players are persons who make logical economic decisions. It is assumed that human people make all of their economic decisions based only on the idea that they are irrational. A player's rewards (utilities, profits, income, or subjective advantages) are assumed to be maximised. [8] The purpose of game-theoretic analysis, when applied to a rational approach, is to provide recommendations on how to make choices against other rational players. First, it reduces the possible outcomes; logical action is more predictable than irrational. Second, it provides a criterion for assessing an economic system's efficiency.

In a Prisoner's Dilemma game between two players, player one and player two can choose the utilities that are the best response to maximise their outcomes. "A best response to a coplayer’s strategy is a strategy that yields the highest payoff against that particular strategy". [9] A matrix is used to present the payoff of both players in the game. For example, the best response of player one is the highest payoff for player one’s move, and vice versa. For player one, they will pick the payoffs from the column strategies. For player two, they will choose their moves based on the two row strategies. Assuming both players do not know the opponents strategies. [10] It is a dominant strategy for the first player to choose a payoff of 5 rather than a payoff of 3 because strategy D is a better response than strategy C.

Applications

Outcome optimisation in game theory has many real world applications that can help predict actions and economic behaviours by other players. [11] Examples of this include stock trades and investments, cost of goods in business, corporate behaviour and even social sciences.[ citation needed ]

Equilibria are not always Pareto efficient, and a number of game theorists design ways to enforce Pareto efficient play, or play that satisfies some other sort of social optimality. The theory of this is called implementation theory.

Related Research Articles

Zero-sum game is a mathematical representation in game theory and economic theory of a situation that involves two sides, where the result is an advantage for one side and an equivalent loss for the other. In other words, player one's gain is equivalent to player two's loss, with the result that the net improvement in benefit of the game is zero.

In game theory, the Nash equilibrium is the most commonly-used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy. The idea of Nash equilibrium dates back to the time of Cournot, who in 1838 applied it to his model of competition in an oligopoly.

The game of chicken, also known as the hawk-dove game or snowdrift game, is a model of conflict for two players in game theory. The principle of the game is that while the ideal outcome is for one player to yield, individuals try to avoid it out of pride, not wanting to look like "chickens." Each player taunts the other to increase the risk of shame in yielding. However, when one player yields, the conflict is avoided, and the game essentially ends.

A coordination game is a type of simultaneous game found in game theory. It describes the situation where a player will earn a higher payoff when they select the same course of action as another player. The game is not one of pure conflict, which results in multiple pure strategy Nash equilibria in which players choose matching strategies. Figure 1 shows a 2-player example.

Matching pennies is a non-cooperative game studied in game theory. It is played between two players, Even and Odd. Each player has a penny and must secretly turn the penny to heads or tails. The players then reveal their choices simultaneously. If the pennies match, then Even wins and keeps both pennies. If the pennies do not match, then Odd wins and keeps both pennies.

In game theory, a player's strategy is any of the options which they choose in a setting where the optimal outcome depends not only on their own actions but on the actions of others. The discipline mainly concerns the action of a player in a game affecting the behavior or actions of other players. Some examples of "games" include chess, bridge, poker, monopoly, diplomacy or battleship. A player's strategy will determine the action which the player will take at any stage of the game. In studying game theory, economists enlist a more rational lens in analyzing decisions rather than the psychological or sociological perspectives taken when analyzing relationships between decisions of two or more parties in different disciplines.

A non-cooperative game is a form of game under the topic of game theory. Non-cooperative games are used in situations where there are competition between the players of the game. In this model, there are no external rules that enforces the cooperation of the players therefore it is typically used to model a competitive environment. This is stated in various accounts most prominent being John Nash's paper.

In game theory, the stag hunt, sometimes referred to as the assurance game, trust dilemma or common interest game, describes a conflict between safety and social cooperation. The stag hunt problem originated with philosopher Jean-Jacques Rousseau in his Discourse on Inequality. In the most common account of this dilemma, which is quite different from Rousseau's, two hunters must decide separately, and without the other knowing, whether to hunt a stag or a hare. However, both hunters know the only way to successfully hunt a stag is with the other's help. One hunter can catch a hare alone with less effort and less time, but it is worth far less than a stag and has much less meat. But both hunters would be better off if both choose the more ambitious and more rewarding goal of getting the stag, giving up some autonomy in exchange for the other hunter's cooperation and added might. This situation is often seen as a useful analogy for many kinds of social cooperation, such as international agreements on climate change.

<span class="mw-page-title-main">Solution concept</span> Formal rule for predicting how a game will be played

In game theory, a solution concept is a formal rule for predicting how a game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players and, therefore, the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium.

Backward induction is the process of determining a sequence of optimal choices by reasoning from the end point of a problem or situation back to its beginning via individual events or actions. Backward induction involves examining the final point in a series of decisions and identifying the most optimal process or action required to arrive at that point. This process continues backward until the best action for every possible point along the sequence is determined. Backward induction was first utilized in 1875 by Arthur Cayley, who discovered the method while attempting to solve the secretary problem.

In game theory, strategic dominance occurs when one strategy is better than another strategy for one player, no matter how that player's opponents may play. Many simple games can be solved using dominance. The opposite, intransitivity, occurs in games where one strategy may be better or worse than another strategy for one player, depending on how the player's opponents may play.

Rationalizability is a solution concept in game theory. It is the most permissive possible solution concept that still requires both players to be at least somewhat rational and know the other players are also rational, i.e. that they do not play dominated strategies. A strategy is rationalizable if there exists some possible set of beliefs both players could have about each other's actions, that would still result in the strategy being played.

In game theory, folk theorems are a class of theorems describing an abundance of Nash equilibrium payoff profiles in repeated games. The original Folk Theorem concerned the payoffs of all the Nash equilibria of an infinitely repeated game. This result was called the Folk Theorem because it was widely known among game theorists in the 1950s, even though no one had published it. Friedman's (1971) Theorem concerns the payoffs of certain subgame-perfect Nash equilibria (SPE) of an infinitely repeated game, and so strengthens the original Folk Theorem by using a stronger equilibrium concept: subgame-perfect Nash equilibria rather than Nash equilibria.

In game theory, a repeated game is an extensive form game that consists of a number of repetitions of some base game. The stage game is usually one of the well-studied 2-person games. Repeated games capture the idea that a player will have to take into account the impact of their current action on the future actions of other players; this impact is sometimes called their reputation. Single stage game or single shot game are names for non-repeated games.

In game theory, a subgame perfect equilibrium is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. Informally, this means that at any point in the game, the players' behavior from that point onward should represent a Nash equilibrium of the continuation game, no matter what happened before. Every finite extensive game with perfect recall has a subgame perfect equilibrium. Perfect recall is a term introduced by Harold W. Kuhn in 1953 and "equivalent to the assertion that each player is allowed by the rules of the game to remember everything he knew at previous moves and all of his choices at those moves".

Risk dominance and payoff dominance are two related refinements of the Nash equilibrium (NE) solution concept in game theory, defined by John Harsanyi and Reinhard Selten. A Nash equilibrium is considered payoff dominant if it is Pareto superior to all other Nash equilibria in the game.1 When faced with a choice among equilibria, all players would agree on the payoff dominant equilibrium since it offers to each player at least as much payoff as the other Nash equilibria. Conversely, a Nash equilibrium is considered risk dominant if it has the largest basin of attraction. This implies that the more uncertainty players have about the actions of the other player(s), the more likely they will choose the strategy corresponding to it.

Equilibrium selection is a concept from game theory which seeks to address reasons for players of a game to select a certain equilibrium over another. The concept is especially relevant in evolutionary game theory, where the different methods of equilibrium selection respond to different ideas of what equilibria will be stable and persistent for one player to play even in the face of deviations of the other players. This is important because there are various equilibrium concepts, and for many particular concepts, such as the Nash equilibrium, many games have multiple equilibria.

In game theory, the traveler's dilemma is a non-zero-sum game in which each player proposes a payoff. The lower of the two proposals wins; the lowball player receives the lowball payoff plus a small bonus, and the highball player receives the same lowball payoff, minus a small penalty. Surprisingly, the Nash equilibrium is for both players to aggressively lowball. The traveler's dilemma is notable in that naive play appears to outperform the Nash equilibrium; this apparent paradox also appears in the centipede game and the finitely-iterated prisoner's dilemma.

In game theory, a strong Nash equilibrium(SNE) is a combination of actions of the different players, in which no coalition of players can cooperatively deviate in a way that strictly benefits all of its members, given that the actions of the other players remain fixed. This is in contrast to simple Nash equilibrium, which considers only deviations by individual players. The concept was introduced by Israel Aumann in 1959. SNE is particularly useful in areas such as the study of voting systems, in which there are typically many more players than possible outcomes, and so plain Nash equilibria are far too abundant.

<span class="mw-page-title-main">Simultaneous game</span>

In game theory, a simultaneous game or static game is a game where each player chooses their action without knowledge of the actions chosen by other players. Simultaneous games contrast with sequential games, which are played by the players taking turns. In other words, both players normally act at the same time in a simultaneous game. Even if the players do not act at the same time, both players are uninformed of each other's move while making their decisions. Normal form representations are usually used for simultaneous games. Given a continuous game, players will have different information sets if the game is simultaneous than if it is sequential because they have less information to act on at each step in the game. For example, in a two player continuous game that is sequential, the second player can act in response to the action taken by the first player. However, this is not possible in a simultaneous game where both players act at the same time.

References

  1. ↑ Osbourne, Martin (2000-11-05). An Introduction to Game Theory (PDF). (Draft). pp. 157–161.
  2. ↑ "Nash Equilibrium: How It Works in Game Theory, Examples, Plus Prisoner's Dilemma". Investopedia. Retrieved 2023-04-23.
  3. ↑ "ICS 180, April 17, 1997". www.ics.uci.edu. Retrieved 2023-04-24.
  4. ↑ "Nash Equilibrium". Corporate Finance Institute. Retrieved 2023-04-23.
  5. ↑ Myerson, Roger B. (1999). "Nash Equilibrium and the History of Economic Theory". Journal of Economic Literature. 37 (3): 1067–1082. doi:10.1257/jel.37.3.1067. ISSN   0022-0515. JSTOR   2564872.
  6. ↑ Wiszniewska-Matyszkiel, Agnieszka (2016-08-01). "Belief distorted Nash equilibria: introduction of a new kind of equilibrium in dynamic games with distorted information". Annals of Operations Research. 243 (1): 147–177. doi: 10.1007/s10479-015-1920-7 . ISSN   1572-9338. S2CID   254235057.
  7. ↑ "What Is the Prisoner's Dilemma and How Does It Work?". Investopedia. Retrieved 2023-04-23.
  8. ↑ Burguillo, Juan C. (2018). Self-organizing coalitions for managing complexity : agent-based simulation of evolutionary game theory models using dynamic social networks for interdisciplinary applications. Cham, Switzerland. ISBN   978-3-319-69896-0.{{cite book}}: CS1 maint: location missing publisher (link)
  9. ↑ Encyclopedia of statistics in behavioral science. Hoboken, N.J.: John Wiley & Sons. 2005. ISBN   978-0-470-86080-9.
  10. ↑ Prisner, E. (2014). Game theory : through examples. [Washington, District of Columbia]. ISBN   978-1-61444-115-1.{{cite book}}: CS1 maint: location missing publisher (link)
  11. ↑ "Game Theory and its Applications". INDUSTRIAL ENGINEERING AND OPERATION RESEARCH. 2019-10-31. Retrieved 2023-04-24.
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