SCIT1003 Chapter 1: Introduction to Game theory Prof. Tsang

Why do we like games? Amusement, thrill and the hope to win Uncertainty – course and outcome of game Escape from the boredom of daily ritual

Reasons for uncertainty randomness combinatorial multiplicity imperfect information

Three types of games bridge

Game Theory 博弈论 Gambling Games of pure luck Chess Combinatorial games

Game Theory 博弈论 Game theory is a study of strategic decision making. Specifically: "the study of mathematical models of conflict and cooperation between intelligent rational decision-makers". Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science, and biology. 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 simplified abstractions of real-world interactions.

What does “game” mean? according to Webster 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)

In a nutshell … Game theory is KM Lecture 4 - Game Theory 9/11/2018 In a nutshell … Game theory is the 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:http://www.ug.bcc.bilkent.edu.tr/~zyilmaz/proposal.html Yale M. Braunstein

Brief History of Game Theory KM Lecture 4 - Game Theory 9/11/2018 Brief History of Game Theory Studies of military strategies dated back to thousands of years ago (Sun Tzu‘s writings孙子兵法) 1913 - E. Zermelo provides the first theorem of game theory; asserts that chess is strictly determined and derived by backward induction 1928 - John von Neumann proves the minimax theorem 1944 - John von Neumann & Oskar Morgenstern write "Theory of Games and Economic Behavior” 1950-1953 - John Nash describes Nash equilibrium (Nobel price 1994) http://william-king.www.drexel.edu/top/class/histf.html Yale M. Braunstein

“Putting yourself in the other person’s shoes” 孙子兵法: 知己知彼 百战不殆 “Putting yourself in the other person’s shoes”

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.

KM Lecture 4 - Game Theory 9/11/2018 Rationality Assumptions: Humans are rational beings They are in the game to win (get rewarded) Humans always seek the best alternative in a set of possible choices Why assume rationality? Narrow down the range of possibilities Predictability Evolution favors those with rational traits 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:http://www.lebow.drexel.edu/economics/mccain/game/game.html Yale M. Braunstein

What’s good for you? Utility Theory KM Lecture 4 - Game Theory 9/11/2018 What’s good for you? Utility Theory Utility Theory based on: rationality maximization of utility (a linear or nonlinear function of income or material reward) Utility (usefulness) is an economic concept, quantifying a personal preference with respect to certain result/reward as oppose to other alternatives. It represents the degree of satisfaction experienced by the player in choosing an action. 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: http://search.britannica.com/bcom/eb/article/5/0,5716,117275+6,00.html Values assigned to alternatives is based on the relative attractiveness to an individual. Yale M. Braunstein

Utility – Example (Exercise) Which would you choose? (Game is only played once!) 10 million Yuan (100% chance), or 100 million Yuan (10% chance) Which would you choose? 10 Yuan (100% chance), or 100 Yuan (10% chance)

What are the “Games” in Game Theory? In Game Theory, our focus is on games where: There are 2 or more players. Where strategy determines player’s choice of action. 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. Examples: Chess, Go, economic markets, politics, elections, family relationships, etc. 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.

KM Lecture 4 - Game Theory 9/11/2018 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. repeated game

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 (strategies). The information that players have available when choosing their actions A description of the payoff (reward) consequences for each player for every possible combination of actions chosen by all players playing the game.

Characteristics of Game Theory The study of decision-making Investigating choices and strategies of actions available to players. It seeks to answer the following questions for the game under consideration: What strategies are available? What kinds of solutions are there? A solution is expressed as a set of strategies for all players that yields a particular payoff, generally the optimal payoff for all players. This payoff is called the value of the game.

Games in real life Games are convenient ways to model strategic interactions among economic agents, or party politics Examples: Behavior in competitive market: e.g. Coca Cola vs. Pepsi Behavior in auctions: bidders bidding against other bidders Behavior in negotiations: e.g. trade negotiations Election campaigns Parliamentary maneuvers

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 (rewards).

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

Zero-Sum & none zero-sum Games KM Lecture 4 - Game Theory 9/11/2018 Zero-Sum & none zero-sum Games When the interests of both sides are in conflict (e.g. chess, sports) the sum of the payoffs remains zero during the course of the game. A game is non-zero-sum, if players interests are not always in direct conflict, so that there are opportunities for both to gain. 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: http://search.britannica.com/bcom/eb/article/5/0,5716,117275+6,00.html Yale M. Braunstein

As a rational game-player, you should Know the payoffs of your actions. Know you opponents’ payoff. Choose the action that maximizes your payoff. Expect your opponents will do the same thing. “Putting yourself in the other person’s shoes”

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

Game Theory in the Real World Economists innovated antitrust policy auctions of radio spectrum licenses for cell phone trade negotiation. Computer scientists new software algorithms and routing protocols Game AI Military strategists nuclear policy and notions of strategic deterrence. Politics Voting, parliamentary maneuver. Biologists How species adopt different strategies to survive, what species have the greatest likelihood of extinction.

Summary: Ch. 1 Essentials of a game Payoffs (Utilities) Normal Form Representation (payoff matrix) Extensive Form Representation (game tree)

Assignment 1.1

Assignment 1.2

Assignment 1.3

Assignment 1.4: draw the game tree for the game “Simple Nim” (Also called the ‘subtraction game’) Rules Two players take turns removing objects from a single heap or pile of objects. On each turn, a player must remove exactly one or two objects. The winner is the one who takes the last object Demonstration: http://education.jlab.org/nim/index.html

Assignment 1.5: Hong Kong Democratic Reform game Demo-parties Accept Reject No reform ? ? Gradual reform One-step reform Central Government

Assignment 1.5: Hong Kong Democratic Reform game The democratic reform process in Hong Kong can be regarded as a 2-player game. On one side is the Central Government in Beijing. On the other side is the democratic parties in the Legislature Council in Hong Kong. According to the Basic Law of the Hong Kong SAR (Special Administrative Region), the Central Government proposes the law for the democratic reform and the democratic parties in Legislature Council can either pass or reject the law. Reform can move forward only if the Central Government proposes the law and the democratic parties in the Legislature Council accept and pass the law. The Central Government can propose law that contains no reform, gradual reform, or one-step (radical) reform, and the democratic parties can accept or reject the law. [a] Assuming the Central Government prefers gradual reform to no reform to radical reform, and the democratic parties prefers radical reform to gradual reform to no reform, you are asked to choose and justify some simple numerical payoffs scheme for this game in normal form. [b] Is this a zero (constant) sum or non- zero (constant) sum game? [c] Is this a cooperative or non-cooperative game? [d] Is this a complete or incomplete information game? [e] Is there a solution to this game if all players are rational? Explain your answers.

Assignment 1.6: Prisoners’ Dilemma- Political contribution reform Existing law in the US allows any amount of political contribution without limit. This allows the rich to buy political influences. But this is difficult to change, because the law-makers are the beneficiaries of this law. Is there any way to convince the law makers in a two-party political system to reform (i.e. pass new law to limit amount of political contribution)? Read p.17-18 (Buffett’s dilemma) in “The Art of Strategy” and draw the payoff matrix for the 2 parties in the game. Explain your answer in detail.