MATH 1020 Chapter 1: Introduction to Game theory

MATH 1020 Chapter 1: Introduction to Game theory
Dr. Tsang

Why do we like games? amusement, thrill and the hope to win
uncertainty – course and result of a game

Reasons for uncertainty
randomness combinatorial multiplicity imperfect information

Three types of games bridge

Game Theory 博弈论 Game theory is the study of how people interact and make decisions. This broad definition applies to most of the social sciences, but game theory applies mathematical models to this interaction under the assumption that each person's behavior impacts the well-being of all other participants in the game. These models are often quite simplified abstractions of real-world interactions.

A cultural comment The Chinese translation “博弈论” may be a bit misleading. Games are serious stuffs in western culture. The Great Game: the strategic rivalry and conflict between the British Empire and the Russian Empire for supremacy in Central Asia ( ). Wargaming: informal name for military simulations, in which theories/tactics of warfare can be tested and refined without the need for actual hostilities.

What does “game” mean? an activity engaged in for diversion or amusement a procedure or strategy for gaining an end a physical or mental competition conducted according to rules with the participants in direct opposition to each other a division of a larger contest any activity undertaken or regarded as a contest involving rivalry, strategy, or struggle <the dating game> <the game of politics> animals under pursuit or taken in hunting

The Great Game: Political cartoon depicting the Afghan Emir Sher Ali with his "friends" the Russian Bear and British Lion (1878)

KM Lecture 4 - Game Theory
12/26/2017 What is Game Theory? Game theory is a study of how to mathematically determine the best strategy for given conditions in order to optimize the outcome “how rational individuals make decisions when they are aware that their actions affect each other and when each individual takes this into account” game theory focuses on how groups of people interact. Game theory focuses on how “players” in economic “games” behave when, to reach their goals, they have to predict how their opponents will react to their moves. CONCLUSION: As a conclusion Game theory is the study of competitive interaction; it analyzes possible outcomes in situations where people are trying to score points off each other, whether in bridge, politics of war. You do this by trying to anticipate the reaction of your competitor to your next move and then factoring that reaction into your actual decision. It teaches people to think several moves ahead. From now on , Whoever it was who said it doesn’t matter if you win or lose but how you play the game, missed the point. It matters very much. According to game theory, it’s how you play the game that usually determines whether you win or lose. Source: Yale M. Braunstein

Brief History of Game Theory
KM Lecture 4 - Game Theory 12/26/2017 Brief History of Game Theory Game theoretic notions go back thousands of years (Sun Tzu‘s writings孙子兵法) E. Zermelo provides the first theorem of game theory; asserts that chess is strictly determined John von Neumann proves the minimax theorem John von Neumann & Oskar Morgenstern write "Theory of Games and Economic Behavior” John Nash describes Nash equilibrium (Nobel price 1994) Yale M. Braunstein

A Beautiful Mind is a 2001 American film based on the life of John Forbes Nash, Jr., a Nobel Laureate in Economics. A Beautiful Mind is an unauthorized biography of Nobel Prize-winning economist and mathematician John Forbes Nash, Jr. by Sylvia Nasar, a New York Times economics correspondent. It inspired the 2001 film by the same name.

12/26/2017 Rationality Assumptions: humans are rational beings humans always seek the best alternative in a set of possible choices Why assume rationality? narrow down the range of possibilities predictability Based on the assumption that human beings are absolutely rational in their economic choices. Specifically, the assumption is that each person maximizes her or his rewards -- profits, incomes, or subjective benefits -- in the circumstances that she or he faces. This hypothesis serves a double purpose in the study of the allocation of resources. First, it narrows the range of possibilities somewhat. Absolutely rational behavior is more predictable than irrational behavior. Second, it provides a criterion for evaluation of the efficiency of an economic system. If the system leads to a reduction in the rewards coming to some people, without producing more than compensating rewards to others (costs greater than benefits, broadly) then something is wrong. Pollution, the overexploitation of fisheries, and inadequate resources committed to research can all be examples of this. Source: Yale M. Braunstein

12/26/2017 Utility Theory Utility Theory based on: rationality maximization of utility may not be a linear function of income or wealth Utility is a quantification of a person's preferences with respect to certain behavior as oppose to other possible ones. In economics, utility is a measure of relative satisfaction. Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one's utility. Utility is often modeled to be affected by consumption of various goods and services, possession of wealth and spending of leisure time. Several subfields have developed along with the analysis of games of strategy. In particular the study of preferences and utility is a virtually independent subject. Daniel Bernouilli (1738, 1954) first suggested that an individual’s subjective valuation of wealth increases at a diminishing rate. Over two hundred years later von Neumann and Morgenstern (1944) provided the precise axioms establishing the existence of a utility function given an individual with a complete preference ordering over a set of riskless options who is permitted to gamble over “lottery tickets” or risky combinations involving the options. Few people would risk a sure gain of $1,000,000 for an even chance of gaining $10,000,000, for example. In fact, many decisions people make, such as buying insurance policies, playing lottery games, and gambling in a casino, indicate that they are not maximizing their average profits. Game theory does not attempt to indicate what a player's goal should be; instead, it shows the player how to attain his goal, whatever it may be. Von Neumann and Morgenstern understood this distinction, and so to accommodate all players, whatever their goals, they constructed a theory of utility. They began by listing certain axioms that they felt all "rational" decision makers would follow (for example, if a person likes tea better than milk, and milk better than coffee, then that person should like tea better than coffee). They then proved that for such rational decision makers it was possible to define a utility function that would reflect an individual's preferences; basically, a utility function assigns to each of a player's alternatives a number that conveys the relative attractiveness of that alternative. Maximizing someone's utility automatically determines his most preferred option. In recent years, however, some doubt has been raised about whether people actually behave in accordance with these rational rules. Source: Values assigned to alternatives is based on the relative attractiveness to an individual. Yale M. Braunstein

Game Theory in the Real World
Economists innovated antitrust policy auctions of radio spectrum licenses for cell phone program that matches medical residents to hospitals. Computer scientists new software algorithms and routing protocols Game AI Military strategists nuclear policy and notions of strategic deterrence. Sports coaching staffs run versus pass or pitch fast balls versus sliders. Biologists what species have the greatest likelihood of extinction.

What are the Games in Game Theory?
For Game Theory, our focus is on games where: There are 2 or more players. There is some choice of action where strategy matters. The game has one or more outcomes, e.g. someone wins, someone loses. The outcome depends on the strategies chosen by all players; there is strategic interaction. What does this rule out? Games of pure chance, e.g. lotteries, slot machines. (Strategies don't matter). Games without strategic interaction between players, e.g. Solitaire. Solitaire, also called Patience, often refers to single-player card games involving a layout of cards with a goal of sorting them in some manner. However it is possible to play the same games competitively (often a head to head race) and cooperatively. The term solitaire is also used for single-player games of concentration and skill using a set layout of tiles, pegs or stones rather than cards. These games include Peg solitaire and Shanghai solitaire.

12/26/2017 Game Theory Finding acceptable, if not optimal, strategies in conflict situations. An abstraction of real complex situation Assumes all human interactions can be understood and navigated by presumptions players are interdependent uncertainty: opponent’s actions are not entirely predictable players take actions to maximize their gain/utilities It is highly mathematical in order to emulated human value judgement (mental rules, fuzzy input of good or bad) ex. Chess play Yale M. Braunstein

Types of games zero-sum or non-zero-sum [if the total payoff of the players is always 0] cooperative or non-cooperative [if players can communicate with each other] complete or incomplete information [if all the players know the same information] two-person or n-person Sequential vs. Simultaneous moves Single Play vs. Iterated

Essential Elements of a Game
The players how many players are there? does nature/chance play a role? A complete description of what the players can do – the set of all possible actions. The information that players have available when choosing their actions A description of the payoff consequences for each player for every possible combination of actions chosen by all players playing the game. A description of all players’ preferences over payoffs.

Normal Form Representation of Games
A common way of representing games, especially simultaneous games, is the normal form representation, which uses a table structure called a payoff matrix to represent the available strategies (or actions) and the payoffs.

A payoff matrix: “to Ad or not to Ad”
PLAYERS Philip Morris No Ad Ad Reynolds 50 , 50 20 , 60 60 , 20 30 , 30 STRATEGIES PAYOFFS

The Prisoners' Dilemma囚徒困境
Two players, prisoners 1, 2. Each prisoner has two possible actions. Prisoner 1: Don't Confess, Confess Prisoner 2: Don't Confess, Confess Players choose actions simultaneously without knowing the action chosen by the other. Payoff consequences quantified in prison years. If neither confesses, each gets 3 year If both confess, each gets 5 years If 1 confesses, he goes free and other gets 10 years Prisoner 1 payoff first, followed by prisoner 2 payoff Payoffs are negative, it is the years of loss of freedom

Prisoners’ Dilemma: payoff matrix
Confess Don’t Confess -5, -5 0, -10 -10, 0 -3, -3 2 1

Prisoner’s Dilemma : Example of Non-Zero Sum Game
A zero-sum game is one in which the players' interests are in direct conflict, e.g. in football, one team wins and the other loses; payoffs sum to zero. A game is non-zero-sum, if players interests are not always in direct conflict, so that there are opportunities for both to gain. For example, when both players choose Don't Confess in the Prisoners' Dilemma

12/26/2017 Zero-Sum Games The sum of the payoffs remains constant during the course of the game. Two sides in conflict Being well informed always helps a player In zero-sum games it never helps a player to give an adversary information, and it never harms a player to learn an opponent's strategy in advance. These rules do not necessarily hold true for nonzero-sum games, however. Source: Yale M. Braunstein

12/26/2017 Non-zero Sum Game The sum of payoffs is not constant during the course of game play. Some nonzero-sum games are positive sum and some are negative sum Players may co-operate or compete. Nonzero-sum game includes all games which are not constant-sum. In non-zero-sum game, the sum of the payoffs are not the same for all outcomes. Nonzero-sum games are mixed motive games. The interests of the players are neither strictly coincident nor strictly opposed. They generate intrapersonal and interpersonal conflicts. They are not always completely soluble but they provide insights into important areas of interdependent choice. In these games, one player's losses do not always equal another player's gains. Some nonzero-sum games are positive sum and some are negative sum: Negative sum games are competitive, but nobody really wins, rather, everybody loses. For example, a war or a strike. Positive sum games are cooperative, all players have one goal that they contribute together as in an educational game. For example, school newspapers or plays, building blocks, or a science exhibit. One major example of a two-person nonzero-sum game is the prisoner's dilemma. It is a non cooperative game because the players can not communicate their intentions. (See topic 'Automata & Games Theory') Source: A player may want his opponent to be well-informed. In a labour-management dispute, for example, if the labour union is prepared for a strike, it behooves it to inform management and thereby possibly achieve its goal without a long, costly conflict. In this example, management is not harmed by the advance information (it, too, benefits by avoiding the costly strike), but in other nonzero-sum games a player can be at a disadvantage if he knows his opponent's strategy. A blackmailer, for example, benefits only if he informs his victim that he will harm the victim unless his terms are met. If he does not give this information to the intended victim, the blackmailer can still do damage but he has no reason to. Thus, knowledge of the blackmailer's strategy works to the victim's disadvantage. Source: Yale M. Braunstein

Information Players have perfect information if they know exactly what has happened every time a decision needs to be made, e.g. in Chess. Otherwise, the game is one of imperfect information.

Imperfect Information
KM Lecture 4 - Game Theory 12/26/2017 Imperfect Information Partial or no information concerning the opponent is given in advance to the player’s decision, e.g. Prisoner’s Dilemma. Imperfect information may be diminished over time if the same game with the same opponent is to be repeated. Yale M. Braunstein

Games of Perfect Information
KM Lecture 4 - Game Theory 12/26/2017 Games of Perfect Information The information concerning an opponent’s move is well known in advance, e.g. chess. All sequential move games are of this type. A class of Game in which players move alternately and each player is completely informed of previous moves. Finite, Zero-Sum, two-player Games with perfect information (including checkers and chess) have a Saddle Point, and therefore one or more optimal strategies. However, the optimal strategy may be so difficult to compute as to be effectively impossible to determine (as in the game of Chess). Source: Yale M. Braunstein

12/26/2017 Games of Co-operation Players may improve payoff through communicating forming binding coalitions & agreements do not apply to zero-sum games Prisoner’s Dilemma with Cooperation Yale M. Braunstein

12/26/2017 Games of Conflict Two sides competing against each other Usually caused by complete lack of information about the opponent or the game Characteristic of zero-sum games Yale M. Braunstein

Example of zero-sum game
Matching Pennies Mis-matcher matcher

Rock-Paper-Scissors

Zero-sum game matrices are sometimes expressed with only one number in each box, in which case each entry is interpreted as a gain for row-player and a loss for column-player.

Strategies A strategy is a “complete plan of action” that fully determines the player's behavior, a decision rule or set of instructions about which actions a player should take following all possible histories up to that stage. The strategy concept is sometimes (wrongly) confused with that of a move. A move is an action taken by a player at some point during the play of a game (e.g., in chess, moving white's Bishop a2 to b3). A strategy on the other hand is a complete algorithm for playing the game, telling a player what to do for every possible situation throughout the game.

Dominant or dominated strategy
A strategy S for a player A is dominant if it is always the best strategies for player A no matter what strategies other players will take. A strategy S for a player A is dominated if it is always one of the worst strategies for player A no matter what strategies other players will take. Even when a player doesn’t have a dominant strategy (i.e., a best strategy, regardless of what the other players do), that player might still have one strategy that dominates another (i.e., a strategy A that is better than strategy B, regardless of what the other players do). As suggested by the terms “best” and “better”, the difference here is between a superlative statement (e.g., “Jane is the best athlete in the class”) and a comparative statement (“Jane is a better athlete than Ted”); because comparatives are weaker statements, we can use them in situations where we might not be able to use superlatives.

If you have a dominant strategy, use it!
KM Lecture 4 - Game Theory 12/26/2017 If you have a dominant strategy, use it! Use strategy 1 Opponent Strategy 1 Strategy 2 Strategy 1 150 1000 You Strategy 2 25 - 10 Yale M. Braunstein

If you have a dominant strategy, use it.
Dominance Solvable If each player has a dominant strategy, the game is dominance solvable COMMANDMENT If you have a dominant strategy, use it. Expect your opponent to use his/her dominant strategy if he/she has one.

Only one player has a Dominant Strategy
The Economist G S Time 100 , 100 0 , 90 95 , 100 95 , 90 For The Economist: G dominant, S dominated Dominated Strategy: There exists another strategy which always does better regardless of opponents’ actions Two firms competing over sales Time and The Economist must decide upon the cover story to run some week. The big stories of the week are: A presidential scandal (labeled S), and A proposal to deploy US forces to Grenada (G) Neither knows which story the other magazine will choose to run

How to recognize a Dominant Strategy
To determine if the row player has any dominant strategy Underline the maximum payoff in each column If the underlined numbers all appear in a row, then it is the dominant strategy for the row player No dominant strategy for the row player in this example.

How to recognize a Dominant Strategy
To determine if the column player has any dominant strategy Underline the maximum payoff in each row If the underlined numbers all appear in a column, then it is the dominant strategy for the column player There is a dominant strategy for the column player in this example.

If there is no dominant strategy
Does any player have a dominant strategy? If there is none, ask “Does any player have a dominated strategy?” If yes, then Eliminate the dominated strategies Reduce the normal-form game Iterate the above procedure

Eliminate any dominated strategy
KM Lecture 4 - Game Theory 12/26/2017 Eliminate any dominated strategy Eliminate strategy 2 as it’s dominated by strategy 1 Opponent Strategy 1 Strategy 2 Strategy 1 150 1000 You Strategy 2 25 - 10 Strategy 3 160 -15 Yale M. Braunstein

Successive Elimination of Dominated Strategies
If a strategy is dominated, eliminate it The size and complexity of the game is reduced Eliminate any dominant strategies from the reduced game Continue doing so successively

Example: Two competing Bars
Two bars (bar 1, bar 2) compete Can charge price of $2, $4, or $5 for a drink 6000 tourists pick a bar randomly 4000 natives select the lowest price bar No dominant strategy for the both players. Bar 2 $2 $4 $5 Bar 1 10 , 10 14 , 12 14 , 15 12 , 14 20 , 20 28 , 15 15 , 14 15 , 28 25 , 25

Successive Elimination of Dominated Strategies
Bar 2 $2 $4 $5 $2 10 , , 10 14 , , 12 14 , , 15 Bar 1 Bar 1 $4 12 , , 14 20 , , 20 28 , , 15 $5 15 , , 14 15 , , 28 25 , , 25 Bar 2 $4 $5 Bar 1 20 , 20 28 , 15 15 , 28 25 , 25

An example for Successive Elimination of strictly dominated strategies, or the process of iterated dominance Question: Does the order of elimination matter? Answer: Although it is not obvious, the end result of iterated strict dominance is always the same regardless of the sequence of eliminations. In other words, if in some game you can either eliminate U for Player 1 or L for Player 2, you don’t need to worry about which one to “do first”: either way you’ll end up at the same answer. If you eliminate a strategy when there is some other strategy that yields payoffs that are higher or equal no matter what the other players do, you are doing iterated weak dominance, and in this case you will not always get the same answer regardless of the sequence of eliminations. (For an example see problem 10.) This is a serious problem, and helps explain why we focus on iterated strict dominance.

Equilibrium The interaction of all players' strategies results in an outcome that we call "equilibrium." Traditional applications of game theory attempt to find equilibria in games. In an equilibrium, each player is playing the strategy that is a "best response" to the strategies of the other players. No one is likely to change his strategy given the strategic choices of the others. Equilibrium is not: The best possible outcome. Equilibrium in the one-shot prisoners' dilemma is for both players to confess. A situation where players always choose the same action. Sometimes equilibrium will involve changing action choices (known as a mixed strategy equilibrium). In game theory, a solution concept is a formal rule for predicting how the game will be played. These predictions are called "solutions", and describe which strategies will be adopted by players, therefore predicting the result of the game. The most commonly used solution concepts are equilibrium concepts, most famously Nash equilibrium. Traditional applications of game theory attempt to find equilibria in these games. In an equilibrium, each player of the game has adopted a strategy that they are unlikely to change. Many equilibrium concepts have been developed (most famously the Nash equilibrium) in an attempt to capture this idea. These equilibrium concepts are motivated differently depending on the field of application, although they often overlap or coincide. This methodology is not without criticism, and debates continue over the appropriateness of particular equilibrium concepts, the appropriateness of equilibria altogether, and the usefulness of mathematical models more generally. An equilibrium s = (s1, , sn) is a strategy profile consisting of a best strategy for each of the n players in the game. The equilibrium strategies are the strategies players pick in trying to maximize their individual payoffs, as distinct from the many possible strategy profiles obtainable by arbitrarily choosing one strategy per player.

Definition: Nash Equilibrium
KM Lecture 4 - Game Theory 12/26/2017 Definition: Nash Equilibrium “If there is a set of strategies with the property that no player can benefit by changing his/her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium.” Source: Nash Equilibrium occurs when the strategies of the various players are best responses to each other. Equivalently but in other words: given the strategies of the other players, each player is acting optimally. Equivalently again: No player can gain by deviating alone, i.e., by changing his or her strategy single-handedly. a unique outcome that satisfied conditions: (1) the solution must be independent of the choice of utility function (if a player prefers x to y and one function assigns x a utility of 10 and y a utility of 1 while a second function assigns them the values 20 and 2, the solution should not change); (2) it must be impossible for both players to simultaneously do better than the Nash solution (a condition known as Pareto optimality); (3) the solution must be independent of irrelevant alternatives (if unattractive options are added to or dropped from the list of alternatives, the Nash solution should not change); and (4) the solution must be symmetrical (if the players reverse their roles, the solution remains the same except that the payoffs are reversed). Source: A set of strategies, one for each player, such that each player’s strategy is best for his/her given that all other players are playing their equilibrium strategies Yale M. Braunstein

Nash equilibrium If each player has chosen a strategy and no player can benefit by changing his/her strategy while the other players keep theirs unchanged, then the current set of strategy choices and the corresponding payoffs constitute a Nash equilibrium.

No strictly dominant strategies and no strictly dominated strategies.
No strictly dominant strategies and no strictly dominated strategies.

Finding Nash equilibria: (a) with strike-outs; (b) with underlinings
A shortcut (but one you should use carefully!) is to underline each player’s best responses.1 To apply this to the game in Figure 11.5, first assume that Player 2 plays L; Player 1’s best response is to play U, so underline the “5” in the box corresponding to (U, L). Next assume that Player 2 plays C; Player 1’s best response is to play D, so underline the “3” in the box corresponding to (D,C). Finally, assume that Player 2 plays R; Player 1’s best response is to play M, so underline the “4” in the box corresponding to (M, R). Now do the same thing for Player 2: go through all of Player 1’s options and underline the best response for Player 2. (Note that C and R are both best responses when Player 1 plays M!) We end up with Figure 11.6b: the only boxes with both payoffs underlined are (D, C) and (M, R), the Nash equilibria of the game.

Prisoner’s Dilemma: finding Dominated Strategies
Which is a Nash Equilibrium?

Prisoner’s Dilemma : Applications
Relevant to: Nuclear arms races. Dispute Resolution and the decision to hire a lawyer. Corruption/political contributions between contractors and politicians. How do players escape this dilemma? Play repeatedly Find a way to ‘guarantee’ cooperation Change payoff structure

Nuclear arms races prisoner's dilemma in disguise
Is there a Nash Equilibrium?

Cigarette Advertising prisoner's dilemma in disguise
Philip Morris No Ad Ad Reynolds 50 , 50 20 , 60 60 , 20 30 , 30

Environmental policy prisoner's dilemma in disguise
Factory C pollution No pollution Factory R 50 , 50 60 , 20 No pollution 20 , 60 20 , 20 Two factories producing same chemical can choose to pollute (lower production cost) or not to pollute (higher production cost).

12/26/2017 Another Example: Big & Little Pigs In the second game, there are two pigs in a long room, one big and one little, and each has two actions. There is a lever at one end of the room, when pushed, gives food at the other end. The Big pig can move the Little pig out of the way and take all the food if they are both at the food output together. The two pigs are equally fast getting across the room, but when they both rush, some of the food, e, is pushed out of the trough and onto the floor where the Little Pig can eat it, and during the time that it takes the Big pig to cross the room, the Little pig can eat of the food. Cost to press button = 2 units When button is pressed, food given = 10 units Yale M. Braunstein

12/26/2017 Decisions, decisions... What’s the best strategy for the little pig? Does he have a dominant strategy? Little Pig Press Wait Press 5 , 1 4 , 4 The Little pig soon figures out that, no matter what Big is doing, Pushing gets nothing but a shock on the (sensitive) snout, so will Wait by the trough. In other words, the strategy Push is dominated by the strategy Wait. Once Little has figured this out, Big will Push, then rush across the room, getting (1 − a) of the food. The unique pure strategy Nash equilibrium is (Wait, Push). Note that the Little pig is getting a really good deal. There are situations where the largest person/firm has the most incentive to provide a public good, and the littler ones have an incentive to free ride. This game gives that in a pure form. Big Pig Wait 9 , -1 0 , 0 Does the big pig have a dominant strategy? Yale M. Braunstein

Research in industries Big & Little Pigs
in disguise Small Company research No research Big Company 5 , 1 4 , 4 No research 9 , -1 0 , 0

Maximin & Minimax Equilibrium in a zero-sum game
KM Lecture 4 - Game Theory 12/26/2017 Maximin & Minimax Equilibrium in a zero-sum game Minimax - minimizing the maximum loss (loss-ceiling, defensive) Maximin - maximizing the minimum gain (gain-floor, offensive) Minimax = Maximin Yale M. Braunstein

Maximin, Minimax & Equilibrium Strategies
KM Lecture 4 - Game Theory 12/26/2017 Maximin, Minimax & Equilibrium Strategies Opponent Strategy 1 Strategy 2 Row Min Strategy 1 150 1000 150 Strategy 2 25 - 10 - 10 You Strategy 3 -15 160 -15 Col. Max 160 1000 Yale M. Braunstein

Saddle point Row Min Col. Max Is this a Nash Equilibrium? 1 3 MaxiMin
4 3 MiniMax A zero-sum game with a saddle point. The outcome of the game depicted in the Figure is (B,R): the maximizer (Rose) wins 3 and the minimizer (Colin) loses 3. Neither player has any incentive to deviate from this outcome. Rather than play B, Rose could play T , but then she would win only 2; rather than play R, Colin could play L, but then he would lose 4 instead of 3. The entry 3 in this matrix game is called a saddle point. The minimax and maximin values of a zero-sum game coincide iff the game has a saddle point, in which case the value of the game is precisely the value of the saddle point. A saddle point in a zero-sum game is an equilibrium in pure actions. It is an action pair from which neither player has any incentive to deviate. But saddle points need not exist.

The Minimax Theorem “Every finite, two-person, zero-sum game
has a rational solution in the form of a pure or mixed strategy.” John Von Neumann, 1926 For every two-person, zero-sum game with finite strategies, there exists a value V and a mixed strategy for each player, such that (a) Given player 2's strategy, the best payoff possible for player 1 is V, and (b) Given player 1's strategy, the best payoff possible for player 2 is −V. Equivalently, Player 1's strategy guarantees him a payoff of V regardless of Player 2's strategy, and similarly Player 2 can guarantee himself a payoff of −V.

Two-Person, Zero-Sum Games: Summary
Represent outcomes as payoffs to row player Find any dominating equilibrium Evaluate row minima and column maxima If maximin=minimax, players adopt pure strategy corresponding to saddle point; choices are in stable equilibrium -- secrecy not required If maximin minimax, find optimal mixed strategy; secrecy essential

Summary: Look for any equilibrium
KM Lecture 4 - Game Theory 12/26/2017 Summary: Look for any equilibrium Dominating Equilibrium Minimax Equilibrium Nash Equilibrium Yale M. Braunstein

Pure & mixed strategies
A pure strategy provides a complete definition of how a player will play a game. It determines the move a player will make for any situation they could face. A mixed strategy is an assignment of a probability to each pure strategy. This allows for a player to randomly select a pure strategy. In a pure strategy a player chooses an action for sure, whereas in a mixed strategy, he chooses a probability distribution over the set of actions available to him.

All you need to know about Probability
If E is an outcome of action, then P(E) denotes the probability that E will occur, with the following properties: 0  P(E)  1 such that: If E can never occur, then P(E) = 0 If E is certain to occur, then P(E) = 1 The probabilities of all the possible outcomes must sum to 1

A zero-sum game: Matching Pennies
Player 2 Player 1 Maximin = minimax : no saddle point No pure Nash Equilibrium: for every pure strategy in this game, one of the players has an incentive to deviate

Mixed strategies for matching pennies
Sticking to a single strategy will not lead to any meaningful solution in matching pennies. So we try a new type of solution: mixing the two choices together. Assume that player 1 picks “Head” with probability p and “Tail” with probability 1-p. If player 2 chooses “H”, he is expected to gain: -p + (1-p) = 1-2p If player 2 chooses “T”, he is expected to gain: +p - (1-p) = -1+2p If player 1 chooses p such that 1 - 2p = p, or p=1/2, then no matter what player 2 does (choosing H or T) he gets the same payoff. Similarly, if player 2 mixes H & T together with probabilities 1/2 & 1/2, then no matter what player 1 does (choosing H or T) he gets the same payoff.

A graphical explanation of Mixed strategies
Payoff for player 2 y(T)=2p-1 1 1 p = probability of choosing H for player 1 -1 y(H)=1-2p Min of the max gain of player 2 = max of min loss of player 1

Another Mixed strategy example
The game “Rock-Paper-Scissors” also do not have a pure strategy equilibrium. In this game, if Player 1 chooses R, Player 2 should choose p, but if Player 2 chooses p, Player 1 should choose S. This continues with Player 2 choosing r in response to the choice S by Player 1, and so forth. In games like Rock-Paper-Scissors, a player will want to randomize over several actions, e.g. he/she can choose R, P & S in equal probabilities. There are a number of motivations given in undergraduate textbooks regarding the need for, and use of, mixed strategies. Pindyck and Rubinfeld (2005) state that,‘there are games… in which a pure strategy is not the best way to play. McCain (2004) describes the need for a mixed strategy when the player would benefit from being ‘unpredictable, so that the opposition cannot guess which strategy is coming and prepare accordingly’. Bernheim and Whinston (2008) support this by motivating their discussion on mixed strategies with‘the key to success… is unpredictability.…The most obvious choice is to make choices randomly. Baldani et al. (2005) write:‘Not all games have pure strategy Nash equilibria in which each player chooses a single strategy with probability one. There are important economic applications that only have solutions in mixed strategies in which a player randomizes by choosing the probabilities for playing the possible pure strategies.’

Mixed strategies no no No Nash equilibrium for pure strategy
x=probability to take action R y=probability to take action S x y 1-x-y 1-x-y=probability to take action P no no No Nash equilibrium for pure strategy

They have to be equal if expected payoff independent of action of player 2

Two-Person, Zero-Sum Game: Mixed Strategies
Column Player: Matrix of Payoffs to Row Player: Row Minima: No dominating strategy A B 1 2 -2 Row Player: Column Maxima: 10 5

Two-Person, Zero-Sum Game: Mixed Strategies
Column Player: Matrix of Payoffs to Row Player: Row Minima: A B 1 2 -2 MaxiMin Row Player: Column Maxima: 10 5 MiniMax MaxiMin MiniMax No Saddle Point!

Optimized Mixed Strategy: Graphical Solution
VR 2A 10 VR < 0*x+10(1-x) 1B VR < 5x-2(1-x)= -2 +7x Optimal Solution: x=12/17, 1-x=5/17 VRMAX=50/17 50/17 If x>12/17, column player can take action A to decrease row player’s expected payoff. If x<12/17, column player can take action B to decrease row player’s expected payoff. 1A 1 x 12/17 Probability of taking action 1 2B

Graphical Solution y= probability of taking action A VR 2A 10 y=1
VR < 10(1-x) y=.75 1B y=0 VR < -2 +7x y=.5 Optimal Solution: x=12/17, 1-x=5/17 VRMAX=50/17 50/17 y=.25 1A 1 x 12/17 2B

x = probability taking action 1

Payoffs of player2 2B 1A Optimal Solution: x=3/7, 1-x=4/7 VRMAX=4+4/7 2A 1B 3/7 1 x Probability of player1 taking action 1

x = probability taking action A
1-x = probability taking action B

Payoff of pure strategy
Payoff of mixed strategy

Pareto optimal Nash equilibrium

An Example of a 3-person non-cooperative game: Truel
N-person games Larger games (More than 2 players) An Example of a 3-person non-cooperative game: Truel

A truel is like a duel, except that three players
A truel is like a duel, except that three players. Each player can either fire, or not fire, his or her gun at either of the other two players. The players’ preferences are: lone survival (the best = 4), survival with another player (the second best = 3), all players’ survival (the second worst=2), the players’ own death (worst case=1). If they have to make their choice simultaneously, what will they do? Ans. All of them will fire at either one of the other two players. If their choices are made sequentially (A>B>C>A>B>…) and the game will continue until only one player survives, what will they do? Ans. They will never shoot.

A don’t shoot shoot C shoot B B B B s C s C ~s ~s s C s A s A ~s s A C ~s s B s A

Example: The paradox of the Chair’s Position
Three voters ABC are electing the chairperson among them. Voter A has 3 votes. Voters B and C have 2 votes each. Voter A’s preference is (ABC). Voter B’s preference is (BCA). Voter C’s preference is (CAB). Who will win if voters vote their first preference? (sincere voting) Who will win if voters will consider what other players may do? (sophisticated voting)

If voters vote sincerely,
Voters A will vote for voter A, voters B will vote for voter B, voters C will vote for voter C. So, the winner is voter A. Let’s consider voters A and BC as follows. A\ (BC) AB BB BC ………. A B C So, the dominant strategy for voter A is voting for A. Assuming voter A will vote for A, let’s consider voters B and C.

B\C A B C So, the dominant strategy for voter C is voting for C. Assuming voters A and C will vote for A and C respectively, let’s consider voter B. B votes for A B C result So, the dominant strategy for voter B is voting for C. As a result, voters A, B and C will vote for A, C and C, respectively. So, the winner is voter C.

Impact of game theory Nash earned the Nobel Prize for economics in 1994 for his “pioneering analysis of equilibria in the theory of non-cooperative games” Nash equilibrium allowed economist Harsanyi to explain “the way that market prices reflect the private information held by market participants” work for which Harsanyi also earned the Nobel Prize for economics in 1994 Psychologist Kahneman earned the Nobel prize for economics in 2002 for “his experiments showing ‘how human decisions may systematically depart from those predicted by standard economic theory’”

Fields affected by Game Theory
Economics and business Philosophy and Ethics Political and military sciences Social science Computer science Biology

MATH 1020 Chapter 1: Introduction to Game theory

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