SCIT1003 Chapter 2: Sequential games - perfect information extensive form Prof. Tsang

Games of Perfect Information KM Lecture 4 - Game Theory 8/26/2018 Games of Perfect Information The information concerning an opponent’s move is well known in advance. Players have perfect information of what has happened every time a decision needs to be made, e.g. chess. All sequential move games are of this type. Otherwise, the game is one of imperfect information. 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:http://mathworld.wolfram.com/PerfectInformation.html Yale M. Braunstein

2- Person Sequential 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 http://en.wikipedia.org/wiki/Nim

Simplified Nim: winning strategy (proof) Lemma: Suppose that Players A and B are playing the Nim subtraction game where at each move a player can remove between 1 and c counters, then a player has a winning strategy if they can play a move that leaves k(c+1) counters. Proof We prove this for Player A (1) Base Case (k=1): Suppose A leaves c+1 counters, then B has to choose to remove x:1≤x≤c. This implies that there are y = c+1-x left, where 1 ≤ y ≤ c. Then A chooses y and wins.

Simplified Nim: proof (2) (2) Inductive step: Assume the statement is true for k=n (n≥1). I.e. if Player A leaves n(c+1) , then player A wins. Suppose A leaves (n+1)(c+1) counters left, i.e. nc+n+c+1 If B chooses x:1≤x≤c, this leaves nc+n+c+1-x. Then A chooses c+1-x, leaving n(c+1). (3) Completion of proof by induction: Thus if the case k=n is true, then so is the case k=n+1 We have the base case k=1, is true, so the statement is true for k=2,3,… and so on. The Lemma is thus proved by induction for all values of k.

Another Example: 21-flags game 21 flags are planted in the field 2 teams (players) Each team moves sequentially and removes 1, 2 or 3 flags each time (0 is not allowed) The team to take the last flag wins.

Game characteristics Sequential: linear chain of moves Perfect information No uncertainty (in probabilistic or strategic sense) Linear chain of reasoning: if I do this he can does that, then I can respond by … Study by drawing a game-tree Game will end at some point (finite game)

Thinking forward At the end, the winning team should leave the other side 4 flags (why?) What should the winning team do before the end? … and before that?

Thinking forward to the end -3 (0) -1 (3) Winner: A -2 (2) B (4) -2 (0) -3 (1) -1 (0)

Thinking forward to before the end -3 (4) -1 (7) Winner: A -2 (6) B (8) -2 (4) -3 (5) -1 (4)

Reasoning backward Whoever (A) moves first will win by removing 1 flag, leaving 20 If B removes 1, A removes 3, leaving 16 If B removes 2, A removes 2, leaving 16 If B removes 3, A removes 1, leaving 16 Next round If B removes 1, A removes 3, leaving 12 If B removes 2, A removes 2, leaving 12 If B removes 3, A removes 1, leaving 12

Otherwise If A removes 3 leaving 18, B removes 2 leaving 16 … At the end A will lose the game

Extensive form: decision/game tree -1 (17) A -2 (16) -1 (20) -1 (18) -3 (15) -2 (19) B -2 (17) A (21) -3 (18) -3 (16)

A Theorem due to E. Zermelo (1871-1951) Binmore (1992, p.32): “Zermelo used this method way back in 1912 to analyze Chess. It requires starting from the end of the game and then working backwards to its beginning. For this reason, the technique is sometimes called ‘backwards induction’.” Binmore, Ken (1992), Fun and Games: A Text on Game Theory, D. C. Heath and Company, Lexington

Rule # 1: Backward reasoning (induction) Think forward Reason backward Next Example: Truel

A truel is like a duel, except that there are three players A truel is like a duel, except that there are three players. Each player has a gun and can either fire, or not fire, his gun at either of the other two players. Each player’s preferences are: Being the lone surviver (best payoff = 4), Survival with another player (the second best = 3), All players survive (the second worst=2), The player’s 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 lift, what will they do? (assume the probability to hit on target is 100%) Ans. They will never shoot.

US Presidential veto game Every year, the US Congress passes bill to authorize expenditures in different areas. President can sign or veto the bill depends on whether he likes it or not. When the Congress and President are controlled by different parties, there is always a flight over how money should be spent, e.g. on M (missile defense) or U (urban renewal). p.41

Which is a better system? With Line-item veto (proposed by Ronald Reagan in 1980s, the president can select the item he opposes to veto), or Without Line-item veto (current system, the president just veto the entire bill) This is a sequential game, with Congress pass the bill first then the President can choose to sign or veto the bill.

Presidential veto game: without Line item veto

Presidential veto game: with Line item veto

Summary: Ch. 2 In study a sequential game, it is useful to analyzing it with a game-tree. Think forward – examining all possible end-game situations. Reason backward – to figure out the correct moves (strategy). [backward induction]

Discussions Is there a first mover advantage in sequential game? Sometimes there is Give an example Sometimes there is not Determine whether order matters in the following examples: Adoption of new technology Class presentation of a project

Assignment 2.1 Analyze the “Century Mark” game described below and figure out a winning strategy: 2 players taking turns At each turn each player chooses an integer between 1 & 10, inclusively, and sum it cumulatively (initial sum = 0) The first player take the cumulative sum greater than or equal to 100 wins the game. Is there a first mover advantage in this game?

Assignment 2.2

Assignment 2.3

Assignment 2.4: Hong Kong Democratic Reform game(in normal form) Demo-parties Accept Reject Gradual reform 1 0 1 -1 One-step reform -1 1 -1 -1 Central Government

Assignment 2.4: Hong Kong Democratic Reform game Express the payoff to both players in a game tree if the Central Gov. is the player that moves first. If the payoff matrix assumed is correct, what would be the result of this game? What is the actual result of this game being played in Hong Kong? Explain why the actual result is different from the expected result.

Assignment 2.5: Tic-Tac-Toe Using a game tree approach to discuss if there is a first mover advantage in this game. Hint: https://www.quora.com/Is-there-a-way-to-never-lose-at-Tic-Tac-Toe