Introduction to Game Theory
Game theory is a separate and interdisciplinary approach for the study of human behavior. The disciplines included in game theory are economics, mathematics and the other social and behavioral sciences. Game theory as like computational theory and so many other contributions was discovered by the great mathematician John von Neumann.
Game theory is a kind of decision theory in which one's alternative action is determined after taking into consideration all possible alternatives available to an opponent playing the similar game, rather than just by the possibilities of various outcome results. Game theory does not insist on how a game must be played but tells the process and principles by which a particular action should be chosen. Therefore it is a decision theory helpful in competitive conditions.
Game is defined as an activity among two or more persons as per a set of rules at the end of which each person gets some benefit or bears loss. The set of rules and procedures defines the game. Going with the set of rules and procedures once by the participants defines the play.
A Scientific Metaphor
As the work of John von Neumann, "games" have been a scientific metaphor for a much diverse range of human interactions in which the outcomes or results depend on the interactive strategies and policies of two or more persons, who have contrast or at best mixed motives. Among the matters discussed in game theory are
1) What does it mean to select strategies "rationally" when outcomes or results depend on the strategies selected by others and when information is partial or incomplete?
2) In "games" that permit mutual gain (or mutual loss) is it "rational" to cooperate to recognize the mutual gain (or avoid the mutual loss) or is it "rational" to do something aggressively in seeking individual gain in spite of of mutual gain or loss?
3) If the answers to 2) are "sometimes," in what situations is aggression rational and in what situations is cooperation rational?
4) In particular, do ongoing relationships different from one-off encounters in this relation?
5) Can moral rules of cooperation arise suddenly from the interactions of rational egoists?
6) How does real human behavior respond to "rational" behavior in such cases?
7) If it differs, then in what direction? Are people tends to be more cooperative than would be "rational?" More aggressive? Or Both?
Therefore, among the "games" considered by game theory are
Why Do Economists Study and Research Games?
- Behavior in imperfectly competitive market, e.g. Pepsi versus Coca-Cola.
- Behavior in auctions, For example- Investment banks bidding on U.S. Treasury bills.
- Behavior in economic negotiations, for example trade.
Properties of a Game
Characteristics of Game Theory
1. Competitive game
A competitive situation is known as competitive game if it has the four properties
The strategy of a player is the determined rule by which player chooses his strategy from his own list during the game. The two types of strategy are
If a player knows precisely what another player is going to do, a deterministic condition is achieved and objective function is to maximize the profit. Thus, the pure strategy is a decision rule always to choose a particular startegy.
If a player is guessing as to which action is to be chosen by the other on any particular instance, a probabilistic condition is achieved and objective function is to maximize the expected profit. Hence the mixed strategy is a choice among pure strategies with fixed probabilities.
Repeated Game Strategies
- In prisoners' dilemma: At start, play doesn't confess. If your opponent plays Confess, then you need to play Confess in the next round. If your opponent plays don't confess, then go for doesn't confess in the subsequent round. This is called as the "tit for tat" strategy.
- In the investment game, if you are sender: At start play Send. Play Send providing the receiver plays Return. If the receiver plays keep, then never go for Send again. This is called as the "grim trigger" strategy.
3. Number of persons
When the number of persons playing is 'n' then the game is known as 'n' person game. The person here means an individual or a group aims at a particular objective.
Two-person, zero-sum game
A game with just two players (player A and player B) is known as 'two-person, zero-sum game', if the losses of one player are equal to the gains of the other one so that the sum total of their net gains or profits is zero.
Two-person, zero-sum games are also known as rectangular games as these are generally presented through a payoff matrix in a rectangular form.
4. Number of activities
The activities can be finite or infinite.
Payoff is referred to as the quantitative measure of satisfaction a person obtains at the end of each play.
6. Payoff matrix
Assume the player A has 'm' activities and the player B has 'n' activities. Then a payoff matrix can be made by accepting the following rules
7. Value of the game
Value of the game is the maximum guaranteed game to player A (maximizing player) when both the players utilizes their best strategies. It is usually signifies with 'V' and it is unique.
Classification of Games
Simultaneous v. Sequential Move Games
- Examples: Sealed-Bid Auctions, Prisoners' Dilemma.
- Must forecast what your opponent will do at this point, finding that your opponent is also doing the same.
- Examples: Bargaining/Negotiations, Chess.
- Must look forward so as to know what action to select now.
- Many sequential move games have deadlines on moves.
One-Shot versus Repeated Games
- Players likely not know much about each another.
- Example - tipping on vacation
- Finitely versus Indefinitely repeated games
- Reputational concerns do matter; opportunities for cooperative behavior may emerge.
Usually games are divided into
The technique for solving these two types changes. By solving a game, we require to determine best strategies for both the players and also to get the value of the game.
Saddle point method can be used to solve pure strategy games.
The diverse methods for solving a mixed strategy game are
Limitations of game theory
The main limitations are
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