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SCIT1003Chapter 1: Introduction to Game theory

Prof. Tsang

Why do we like games?

• Amusement, thrill and the hope to win• Uncertainty – course and result of a game

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Reasons for uncertainty• randomness• combinatorial multiplicity• imperfect information

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Three types of games

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bridge

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Game Theory 博弈论

GamblingGames of pure luck

ChessCombinatorial 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 quite simplified abstractions of real-world interactions. 7

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 hunting8

The Great Game:

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

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In a nutshell … Game theory isthe 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”

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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

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)

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孙子兵法 :

知己知彼 百战不殆

“Putting yourself in the other person’s shoes”

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RationalityAssumptions: • Humans are rational beings• They are in the game to win (get rewarded)• Humans always seek the best alternative in

a set of possible choicesWhy assume rationality?• narrow down the range of possibilities• predictability

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What’s good for you?Utility Theory

Utility Theory based on:• rationality• maximization of utility– may not be a linear function of income or material

reward

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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.

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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.

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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

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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

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Essential Elements of a Game1. The players

• how many players are there? • does nature/chance play a role?

2. A complete description of what the players can do – the set of all possible actions (strategies).

3. The information that players have available when choosing their actions

4. A description of the payoff (reward) consequences for each player for every possible combination of actions chosen by all players playing the game.

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• Fundamentally about the study of decision-making• Investigations are concerned with choices and

strategies of actions available to players.

It seeks to answer the questions: i. What strategies are there?ii. 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.

Characteristics of Game Theory

Games & economics• Games are convenient ways to model strategic

interactions among economic agents.• Many economic situations involve strategic

interactions– 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

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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).

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A payoff matrix: “to Ad or not to Ad”

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PLAYERS

STRATEGIESPAYOFFS

Philip Morris

No Ad Ad

Reynolds No Ad 50 , 50 20 , 60

Ad 60 , 20 30 , 30

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

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Prisoners’ Dilemma: payoff matrix

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Confess Don’t Confess

Confess -5, -5 0, -10

Don’t Confess

-10, 0 -3, -3

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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.

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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”

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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’”

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Fields affected by Game Theory

• Economics and business• Philosophy and Ethics• Political and military sciences• Social science• Computer science• Biology

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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.

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Summary: Ch. 1

• Essentials of a game• Payoffs (Utilities)• Normal Form Representation (payoff matrix)• Extensive Form Representation (game tree)

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Assignment 1.1

Assignment 1.2

Assignment 1.3

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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 objectDemonstration: http://education.jlab.org/nim/index.html

Assignment 1.5: Hong Kong Democratic Reform game

Accept Reject

No reform ? ? ? ?

Gradual reform

? ? ? ?

One-step reform

? ? ? ?

Demo-parties

Central Government

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, choose and justify some simple numerical payoffs 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 1.5: Hong Kong Democratic Reform game