ncp1 intro

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Game Theory

Lecture Notes By

Y. NarahariDepartment of Computer Science and Automation

Indian Institute of Science

Bangalore, India

February 2008

Chapter 1: Introduction to Game Theory

Note: This is a only a draft version, so there could be flaws. If you find any errors, please do send email to

[email protected]. A more thorough version would be available soon in this space.

• According to Myerson [1], game theory may be defined as the study of mathematical models of conflict and cooperation between rational, intelligent decision makers.

• Game theory provides general mathematical techniques for analyzing situations in which two ormore individuals (called players or agents) make decisions that influence one another’s welfare.

• More appropriate phrases to describe game theory are:

– Conflict Analysis

– Interactive Decision Theory• According to Osborne and Rubinstein [2], a game is a description of strategic interaction that

occurs among decision-making entities.

A game can be described by four elements according to Mascollel, Whinston, and Green [3]:

1. Players (also called agents): an individual or a group of individuals making a decision

2. Rules specify

• who moves when

• what can they do• what do they know when they move

3. Outcomes: What happens when each player plays in a particular way?

4. Preferences: What are the players’ preferences over the possible outcomes.

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Example 1: Matching Pennies

(1) Players: {1,2}

(2) Rules: Each player simultaneously puts down a coin, heads up or tails up

(3) Outcomes: {(H,H), (H,T), (T,H), (T,T)}

(4) Preferences: If the two coins match, player 1 pays Rs. 100 to player 2; otherwise player 2 paysRs. 100 to player 1.

• This game is called the Matching Pennies game.

Example 2: Tick-Tack-Toe

(1) Players {X, O}

(2) Rules

– There is a board that consists of 9 squares in 3 rows and 3 columns

– Players take turns putting their marks (either X or O) into an unmarked square.

– Player X moves first

– The players observe all the choices made by the players so far.

(3) Outcomes: win , loss , or draw

– The first player to have 3 marks in a horizontal row or a vertical column or along thediagonal wins and the other one loses

– If no one succeeds even after all 9 squares are marked, it is a draw.

(4) Preferences

– The player who wins will receive a certain amount of money from the player who loses

– No payments are made if it is a draw.

Figure 1 shows a sequence of moves in a tick-tack-toe game.

Important Assumptions and Issues in Game Theory

Utility Theory

Utility theory enables the preferences of the players to expressed in terms of payoffs in some utilityscale. Utility theory is the science of assigning numbers to outcomes in a way that reflects a player’spreferences. Utility theory is one of the landmark contributions of von Neumann and Morgenstern.They stated and proved in [4], a crucial theorem called the expected utility maximization theorem .This theorem shows for any rational decision maker that there must exist some way of assigningutility numbers to different outcomes in a way that the decision maker would always choose the option

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X OX X

X

XX

XX

X XX

X

X

X

X

X

X

O

O

O

OO

O

O

O

O

O

O

loses

wins

O

Figure 1: A sequence of moves in Tick-Tack-Toe

that maximizes his expected utility. This theorem holds under fairly weak assumptions about how arational decision maker should b ehave. For example, one of the key assumptions is the sure-thing orthe substitution axiom. This axiom informally means the following: If a decision maker prefers option1 over option 2 when event A occurs and he prefers option 1 over option 2 when event A does not

occur, then he should prefer option 1 over option 2 even before learning whether event A will occuror not. Using the above assumption and certain technical regularity conditions, von Neumann andMorgenstern showed that there exists some utility scale such that the decision maker always prefersthe options that give him the highest expected value.

Rationality

The players are assumed to be rational .

• A decision maker is said to be rational if the agent makes decisions consistently in pursuit of hisown objectives. It is assumed that each player’s ob jective is to maximize the expected value of his own payoff measured in some utility scale. This is the selfish part of rationality. The above

notion of rationality (maximization of expected utility payoff) is attributed to Bernoulli (1738)and von Neumann and Morgenstern (1944) [4].

• It is important to note that maximizing expected utility payoff is not necessarily the same asmaximizing expected monetary payoff. For example, a certain amount of money may be of significant utility to a person desperately in need of the money. The same amount of money maybe of insignificant utility to a person who does not need it because he is already rich. In general

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utility and monetary payoff are non-linearly related. Specifically, it depends on the nature of the agent: risk-loving or risk-averse or risk-neutral.

• When there are two or more decision makers, it would be the case that the rational solution toeach individual’s decision problem depends on the others’ individual problems and vice-versa.None of the problems may be solvable without understanding the solutions of the other problems.

When such rational decision makers interact, their decision problems must be analyzed together,like a system of simultaneous equations. Game theory precisely deals with such analysis.

• It is to be noted that selfishness or self-interest is an important implication of rationality.

Intelligence

The players are assumed to be intelligent.

• This assumption means that each player in the game knows everything about the game thata game theorist knows and the player can make any inferences about the game that a gametheorist can make.

• In particular, an intelligent player is strategic , that is, fully takes into account his knowledge orexpectation of behavior of other agents in figuring out what his best response strategy should be.Each player has enough computational resources to do any required computations for determininghis best response strategy.

Justification for Rationality and Intelligence Assumptions

Myerson provides a very convincing explanation to show that the assumptions of rationality andintelligence are reasonable. According to him, the assumption that all individuals are rational andintelligent may not exactly be satisfied in a typical real-world situation. However, any theory that isnot consistent with the assumptions or rationality and intelligence is fallible for the following reason.

If a theory makes the prediction that some individuals will be systematically fooled or misled intomaking mistakes, then such a theory will tend to lose validity when individuals learn through mistakesto better understand the situations. On the other hand, a theory based on rationality and intelligenceassumptions is credible and will stand the test of time due to its consistency.

Common Knowledge

This is an important implication of intelligence . Aumann (1976) defined common knowledge as follows:A fact is common knowledge among the players if every player knows it, every player knows that everyplayer knows it, and so on. That is, every statement of the form ” ( every player knows that) k everyplayer knows it” is true for k = 0, 1, 2, . . ..

The intelligence assumption means that whatever a game theorist may know or understand aboutthe game must be known or understood by the players of the game. Thus the model of the game isalso known to the players. Since all the players know the model and they are intelligent, they alsoknow that they all know the model. Thus the model is common knowledge.

A player’s private information is any information that he has that is not common knowledge amongall the players.

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An Example for Illustrating Common Knowledge

This example is a variation of the one presented by Myerson [1]. Assume that there are five rationaland intelligent mothers A, B, C, D, and E and let a, b, c, d, and e be their sons, respectively. Thekids go the school every day, escorted by their respective mothers and the mothers get an opportunityeveryday to indulge in some conservation. The conversation invariably centers around the performance

and behavior of the kids.

• Everyday when the 5 mothers meet, the conversation protocol is the following. If a motherthinks her boy is good boy , she will praise the virtues of her boy. On the other hand, if a motherknows that her son is not a good boy , then she will cry. All mothers follow this protocol.

• None of the boys is a good boy but they are smart in that their bad ways are not known to theirrespective mothers. Whenever a mother finds that the kid of another mother badly behaved, shewould immediately report it to all mothers except the kid’s mother. For example, if A finds bbadly behaved, then A would report it to C,D, and E, but not to B. This protocol is also knownto all the mothers. Since all the kids are badly behaved, the fact that a kid is not well behavedis common knowledge among all the mothers except the kid’s own mother.

• Since each mother does not know that her boy is badly behaved, it turns out that every motherkeeps praising her son everyday.

• One fine day, the class teacher meets all the mothers and makes the following statement: ”oneof the boys is a bad boy.” Thus the fact that one of the boys is not a good boy is commonknowledge among all the mothers.

• Thereafter, when the five mothers meet the next day, all of them praise their respective boys;the same happens on the 2nd day, 3rd, and the 4th day.

• On the 5th day, however, all the mothers cry together because all of them realize that their

respective sons are bad boys.

• Note that the announcement made by the class teacher is common knowledge and that is whatmakes all the mothers cry on the fifth day.

Bounded Rationality

Osborne and Rubinstein [2] provide the following explanation regarding the important notion of bounded rationality.

• Game theory, in its existing form, assumes that all the players are symmetric; that is, they haveidentical capabilities of perception and computation. It does not model asymmetries in abilitiesor perceptions of situations.

• An outstanding example of this is the game of chess. When analyzed using game theory, thegame of chess can be solved using an algorithm. This was in fact shown by Zermelo in 1913.Thus chess becomes a trivial game for rational players.

• Zermelo’s Result was that the game of chess has a unique equilibrium outcome with the propertythat a player who follows the suggested strategy can be sure that the outcome will be at least as

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good as the equilibrium outcome. Only the existence of the equilibrium outcome has been shownand the equilibrium outcome is yet to be computed. A game theoretic model of chess revealstherefore reveals an important fact about the game and suggests that it is not at all interesting forrational players. However, the game theoretic model does not capture the asymmetric abilitiesof the players which is what makes it interesting.

Bounded rationality grapples with the problem of asymmetric abilities of the players and the fact thatthe players do not have infinite computational resources at their disposal.

Classification of Games

Non-cooperative Games and Cooperative Games

Non-cooperative games are those in which the possible actions of individual players are the primitives;in cooperative games, joint actions of groups of players are the primitives. John Harsanyi (1966)explained that a game is cooperative if commitments (agreements, promises, threats) are enforceableand that a game becomes non-cooperative if the commitments are not enforceable.

Different Representational Forms

• A strategic form game (also called simultaneous move game or normal from game) is a model ora situation where each player chooses his plan of action once and for all and all players exercisetheir decision simultaneously.

• An extensive form game specifies a possible order of events and each player can consider his planof action whenever a decision has to be made.

• A coalitional form game or characteristic form game is one where every subset of players of playersis represented, by associating a value with each subset of players. This form is appropriate forcooperative games.

Perfect Information and Games with Imperfect Information

When the players are fully informed about each other’s moves (each player, before making a move,knows the past moves of all other players as well as his own past moves), the game is said to be of perfect information. Otherwise the game is said to be with imperfect information.

Complete Information and Incomplete Information

A game with incomplete information is one in which, at the first point in time when the players canbegin to plan their moves, some players already have private information about the game that otherplayers do not know. In a game with complete information, every aspect of the game is common

knowledge.

Other Categorizations

There are many other categories of games, such as dynamic games, repeated games, stochastic games,games with communication, multi-level games (Stackelberg games), differential games, etc. These willbe discussed in due course of time.

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To Probe Further

The material discussed in this chapter draws upon mainly from three sources, namely the books byMyerson [1], Mascolell, Whinston, and Green [3], and Osborne and Rubinstein [2].

The classic treatise by John von Neumann and Oskar Morgenstern [4], published in 1944, provides acomprehensive foundation for game theory. To this day, even after seven decades of its first appearance,

the book continues to be a valuable reference.The first chapter of Myerson’s book [1] presents a detailed treatment of decision theory and utility

theory.For an undergraduate level treatment of game theory, we recommend the books by Osborne [5],

Straffin [6], and Binmore [7]. For a graduate level treatment, we recommend the books by Myerson[1] and Osborne and Rubinstein [2].

References

[1] Roger B. Myerson. Game Theory: Analysis of Conflict . Harvard University Press, 1997.

[2] Martin J. Osborne and Ariel Rubinstein. A Course in Game Theory . Oxford University Press,1994.

[3] Andreu Mas-Collel, Michael D. Whinston, and Jerry R. Green. Micoreconomic Theory . OxfordUniversity Press, 1995.

[4] John von Neumann and Oskar Morgenstern. Theory of Games and Economic Behavior . PrincetonUniversity Press, 1944.

[5] Martin J. Osborne. An Introduction to Game Theory . The MIT Press, 2003.

[6] Philip D. Straffin Jr. Game Theory and Strategy . The Mathematical Association of America, 1993.

[7] Ken Binmore. Fun and Games : A Text On Game Theory . D. C. Heath & Company, 1992.

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