Decisions and Games. Lunch Lecture for W.I.S.V. Christiaan Huygens at TU Delft 28 November Tom Verhoeff
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From this document you will learn the answers to the following questions:
Who is the name of the person who created the game?
What is the maximum payoff for Alice?
What is the term for the Nash Equilibrium?
Transcription
1 Decisions and Games Lunch Lecture for W.I.S.V. Christiaan Huygens at TU Delft 28 November 2008 Tom Verhoeff Department of Mathematics & Computer Science c 2008, T. TUE.NL /9 Decisions and Games
2 First Game: Choosing Digits Two players: Choose a digit, taking turns. Earlier chosen digits are no longer available. Goal: Having chosen three digits that sum to 5. c 2008, T. TUE.NL 2/9 Decisions and Games
3 Logical Combinations Analysis can be extremely hard! c 2008, T. TUE.NL 3/9 Decisions and Games
4 Second Game: Choosing Heads or Tails Bob chooses 0 Alice chooses Payoff Two players: Each chooses 0 (head) or (tail), secret the other. Goal: Maximize total payoff under repeated play. c 2008, T. TUE.NL 4/9 Decisions and Games
5 Strategic Bluffing: The Mixed Strategy Analysis for Alice: she chooses with probability x%. Alice wants to determine optimal x, not knowing Bob s choice. 2 3 Verwachte opbrengst als Bob 0 kiest x 2 3 Verwachte opbrengst als Bob kiest x Expected payoff = +0, 5 Expected payoff = 0, 5 voor x = 50% voor x = 50% c 2008, T. TUE.NL 5/9 Decisions and Games
6 Strategic Bluffing: The Nash Equilibrium 2 Verwachte opbrengst als Bob optimaal kiest rood Expected Payoff for Alice x Alice x Bob y 0.25 Optimal for Alice: choose 0 with probability 5/8 & choose with probability 3/8 Expected payoff for Alice: +0,25 or 6 4 % of 2 Moreover, this is independent of Bob s choice. c 2008, T. TUE.NL 6/9 Decisions and Games
7 The Effect of Variance Expected payoff µ = 0, 25 with standard deviation of σ =, 884 Three times 000 experiments of N repetitions each N = 20 N = 200 N = σ = 0, 42 σ = 0, 3 σ = 0, 04 38% chance to lose 7% chance to lose 0, % chance to lose c 2008, T. TUE.NL 7/9 Decisions and Games
8 Third Game: Choosing Categories Category Score Double (value + value)... Square (value value)... Total... One player: Roll dice, choose unscored category, repeat. Goal: Maximize total score under repeated play. c 2008, T. TUE.NL 8/9 Decisions and Games
9 Weighing Chances: MicroYahtzee Game Graph Micro Yahtzee: spelgraaf Begin 2 3 Dubbel Kwadraat Einde Kwadraat Dubbel Werp Kies Werp "Kies" Copyright (c) , Tom Verhoeff c 2008, T. TUE.NL 9/9 Decisions and Games
10 Weighing Chances: MicroYahtzee Expected Scores Micro Yahtzee: Verwachte scores Werp Kies Werp "Kies" Copyright (c) , Tom Verhoeff c 2008, T. TUE.NL 0/9 Decisions and Games
11 Weighing Chances: MicroYahtzee Optimal Scores Micro Yahtzee: Optimaal verwachte score / Werp Kies Werp 2 "Kies" Copyright (c) , Tom Verhoeff c 2008, T. TUE.NL /9 Decisions and Games
12 Weighing Chances: MicroYahtzee Dilemma Resolved Micro Yahtzee: Optimaal verwachte score 7 / /6 9 / / / / Werp Kies Werp 2 "Kies" Copyright (c) , Tom Verhoeff c 2008, T. TUE.NL 2/9 Decisions and Games
13 Weighing Chances: Real Yahtzee Score card with 3 categories and 5 dice Choose 39 : more than 0 9 game states Expected total score under optimal play: 254, [Verhoeff, 999] But variance is high: σ = ±60 (70% in ) On-line advice and practicing: c 2008, T. TUE.NL 3/9 Decisions and Games
14 Optimal Solitaire Yahtzee: Final Scores per Category Category E SD % 0 Aces Twos Threes Fours Fives Sixes U. S. Bonus Three of a Kind Four of a Kind Full House Small Straight Large Straight Yahtzee Chance Extra Y. Bonus GRAND TOTAL Yahtzees Rolled Jokers Applied c 2008, T. TUE.NL 4/9 Decisions and Games
15 Optimal Solitaire Yahtzee: Distribution of Final Score Score range % Cum.% % 0 % % 0 % % 2 % % 5 % % 4 % % 27 % % 4 % % 60 % % 80 % % 86 % % 90 % % 92 % % 93 % % 94 % % 96 % % 98 % % 99 % % 99 % % 99 % % 99 % Results based on simulation of 0 5 games c 2008, T. TUE.NL 5/9 Decisions and Games
16 Optimal Solitaire Yahtzee: Game with Minimum Score Turn Third Roll Score in Category Aces Twos Four of a Kind Yahtzee Threes Fours Fives Full House Sixes Large Straight Chance Three of a Kind Small Straight 2 GRAND TOTAL c 2008, T. TUE.NL 6/9 Decisions and Games
17 Three Kinds of Situations for Making Decisions Logical combinations (complete information) Strategic bluffing (secrets) Weighing chances (fortune) c 2008, T. TUE.NL 7/9 Decisions and Games
18 Wise Lessons. Playful mathematical techniques may help at making good choices in diverse situations. 2. For repeated strategical choices it can be optimal to toss a (well-chosen) coin (cf. the so-called Mixed Nash Strategy, preventing predictability and exploitation). 3. For repeated tests of fortune it can be optimal to make a (well-chosen) fixed choice (cf. the so-called Markov Decision Processes). 4. Large variance requires patience for increased certainty (due to the factor / N). c 2008, T. TUE.NL 8/9 Decisions and Games
19 Decide Playfully, Consult a Mathematician! c 2006, 9 Decisions and Games
20 Literature Jörg Bewersdorff. Glück, Logik und Bluff. (Eng. translation: Luck, Logic, and White Lies.) Berlekamp, Conway, Guy. Winning Ways for Your Mathematical Plays. (4 volumes) en.wikipedia.org/wiki/winning_ways_for_your_mathematical_plays William Poundstone. Prisoner s Dilemma. en.wikipedia.org/wiki/prisoner%27s_dilemma c 2006, 20 Decisions and Games
TomVerhoeff. Technische Universiteit Eindhoven Faculty of Math. & Computing Science Software Construction
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