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

• Bankruptcy
• Caveat Emptor
• Barbarians at the Gate
• Battle of the Networks
• Escape and Evasion
• Conscription
• Coordination
• Majority Rule
• Hawk versus Dove
• Mutually Assured Destruction
• Market Niche
• Mutual Defense
• Ultimatum
• Subsidized Small Business
• Tragedy of the Commons
• Frogs Call for Mates
• Prisoner's Dilemma
• Video System Coordination

Why Do Economists Study and Research Games?

• Games are a suitable way to model the strategic interactions among economic agents.
• Many economic topics include strategic interaction.

-        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.

• Game theory is not restricted to Economics

 Properties of a Game

1. There are finite number of competitors known as 'players'
2. All the strategies and their impacts are specified to the players but player does not know which strategy is to be selected.
3. Each player has a limited number of possible courses of action known as 'strategies'
4. A game is played when every player selects one of his strategies. The strategies are supposed to be prepared simultaneously with an outcome such that no player recognizes his opponent's strategy until he chooses his own strategy.
5. The figures present as the outcomes of strategies in a matrix form are known as 'pay-off matrix'.
6. The game is a blend of the strategies and in certain units which finds out the gain or loss.
7. The player playing the game always attempts to select the best course of action which results in optimal pay off known as 'optimal strategy'.
8. The expected pay off when all the players of the game go after their optimal strategies is called as 'value of the game'. The main aim of a problem of a game is to determine the value of the game.
9. The game is said to be 'fair' if the value of the game is zero or else it s known as 'unfair'.

Characteristics of Game Theory

1. Competitive game

A competitive situation is known as competitive game if it has the four properties

1. There are limited number of competitors such that n ≥ 2. In the case of n = 2, it is known as two-person game and in case of n > 2, it is known as n-person game.
2. Each player has a record of finite number of possible actions.
3. A play is said to takes place when each player selects one of his activities. The choices are supposed to be made simultaneously i.e. no player knows the selection of the other until he has chosen on his own.
4. Every combination of activities finds out an outcome which results in a gain of payments to every player, provided each player is playing openly to get as much as possible. Negative gain means the loss of same amount.

2. Strategy

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

1. Pure strategy
2. Mixed strategy

Pure Strategy

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.

Mixed Strategy

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 repeated games, the chronological nature of the relationship permits for the acceptance of strategies that are dependent on the actions chosen in previous plays of the game.
• Most contingent strategies are of the kind called as "trigger" strategies.
• For Example trigger 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.

5. Payoff

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

• Row designations for every matrix are the activities or actions available to player A
• Column designations for every matrix are the activities or actions available to player B
• Cell entry V_ij is the payment to player A in A's payoff matrix when A selects the activity i and B selects the activity j.
• In a zero-sum, two-person game, the cell entry in the player B's payoff matrix will be negative of the related cell entry V_ij in the player A's payoff matrix in order that total sum of payoff matrices for player A and player B is finally zero.

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

• Games where players select activities simultaneously are simultaneous 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.

• Games where players select activities in a particular series or sequence are sequential move games.

-        Examples: Bargaining/Negotiations, Chess.

-        Must look forward so as to know what action to select now.

-        Many sequential move games have deadlines on moves.

• Many strategic situations include both sequential and simultaneous moves.

One-Shot versus Repeated Games

• One-shot: play of the game takes place once.

-        Players likely not know much about each another.

-        Example - tipping on vacation

• Repeated: play of the game is recurring with the same players.

-        Finitely versus Indefinitely repeated games

-        Reputational concerns do matter; opportunities for cooperative behavior may emerge.

• Advise: If you plan to follow an aggressive strategy, ask yourself whether you are in a one-shot game or in repeated game. If a repeated game then think again.

Usually games are divided into

• Pure strategy games
• Mixed strategy games

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

• Dominance rule
• Analytical method
• Graphical method
• Simplex method

Limitations of game theory

The main limitations are

• The hypothesis that the players have the information about their own payoffs and others is rather impractical
• As the number of players adds in the game, the analysis of the gaming strategies turns out to be increasingly intricate and complicated.
• The assumptions of maximin and minimax presents that the players are risk-averse and have whole information of the strategies. It doesn't look practical.
• Rather than each player in an oligopoly condition working under uncertain situations, the players will permit each other to share the secrets of business so as to work out collusion. Then the mixed strategies are not very helpful.