# WONG TO YEUNG LEO WONG KA WANG KELVIN LAI CHUNG YIN ROY How to play games?

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WONG TO YEUNG LEO WONG KA WANG KELVIN LAI CHUNG YIN ROY How to play games?

Are these game? Paper-scissor-stone Chinese chess Penalty kick Voting Government deciding if it should give tax rebate All are games !

More about game PlayerSSSS - at least two people - their choice will affect each other Strategies - action that players can choose Payoff - ranking of the outcome by each players

Example Paper-scissor-stone game Number of players = 2 Strategies: paper, scissor, stone Payoff: 1 for winning ( ∵ happy =]) 0 for draw -1 for losing ( ∵ unhappy =[ )

Classification of game Sequential game - players take turn to play e.g. playing Chinese chess Simultaneous game - players play at the same time e.g. paper-scissor-stone

How to solve sequential game? Solve: Determine the strategy each players will choose Principle: Look forward, reason backward. Backward reasoning

Example 1 (Divide-and-choose) Situation: A mother buys a cake for her son and daughter, who love cakes so much ! Rule : the daughter cuts the cake first, then the son chooses the part he wants to have.

Example 1 (Divide-and-choose) Assume the daughter has two strategies: 1, cut it into half 2, cut it into one-fourth and three-fourth The son has two strategies: 1, choose the larger one 2, choose the smaller one

Example 1 (Divide-and-choose)

Solution: Daughter: cut in a half Son: large or small ( just the same)

Example 2 (sailing competition) Situation: Team A is leading Team B in a sailing competition Rule: Team B chooses an action in the hope to surpass Team A, then Team A responds

Example 2 (sailing competition) Team B has two strategies: 1, go left 2, go right Them A has two strategies: 1, follow Team B 2, Not follow Team B

Example 2 (sailing competition)

Solution: Team B: go right Team A: follows B

Example 3 (grading scheme) Situation: In MATH 162, students have the chance to give marks to other students, which will affect the course grade. Rule: Prof. Li chooses the grading scheme first. Students then choose whether they will intentionally give low marks.

Example 3 (grading scheme) Prof. Li has two strategies: 1. grading by curve 2. absolute grading Student has two strategies: 1. always give low marks 2. give marks sincerely

Example 3 (grading scheme)

Solution: Prof. Li: absolute grading Student: give marks sincerely We get the right scheme !

Let’s play a game! I have a 10 dollars notes The first player proposes a method on how to divide it. e.g. 1, I get all 10 dollars, you get nothing I give you all 10 dollars, I get nothing

Let’s play a game! The second player can choose whether to accept or not. - If he accepts, we will divide the 10 dollars note as proposed by first player. - If he turns down, none of them get anything.

Let’s play a game! On a paper, write down: 1, if you are player 1, what will you propose. 2, if you are player 2, the minimum amount that you need to get in order to accept player 1‘s offer

Let’s play a game! What should be the solution? Famous game: Ultimatum game

For interested students….

Homework 1, Solve the divide-and-choose game when the daughter can choose to divide in any proportion she likes. You can consider the two strategies to be: divide equally and divide unequally 2, Solve the Ultimatum game. You can consider the strategies of player 1 to be: player 1 get all, player 1 get \$9.9 and player 1 get less than \$9.9

Homework (Extra) Andy, Ben, Charles are in a one-round three-way duel. They take turn shoot at each other or fire up in the air. They never miss a shoot. Andy start first, followed by Ben and Charles. They want to be the sole survivor and never want to die. Show that Andy will fire up in the air. [Hint: backward reasoning makes things complicated this time]

Discussion 1, Do you find game theory interesting? 2, Have you used backward reasoning in daily life? 3, In Ultimatum game, players usually do not play the solution at first. Can you suggest some reasons?

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