T-79.4001 The Hat Problem

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A seminar presentation given on the TKK course T-79.4001.

Describes the hat problem, which combines information theory and game theory in a novel way.

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T-79.4001 The Hat Problem

  1. 1. Outline The Problem The Solution Variants T-79.4001: The Hat Problem Janne Peltola 5.3.2008 Janne Peltola T-79.4001: The Hat Problem
  2. 2. Outline The Problem The Solution Variants 1 The Problem Introduction Solutions 2 The Solution Algorithm Efficiency 3 Variants Janne Peltola T-79.4001: The Hat Problem
  3. 3. Outline The Problem The Solution Variants The situation n players play a game. The rules are simple. The players enter a room blindfolded. Janne Peltola T-79.4001: The Hat Problem
  4. 4. Outline The Problem The Solution Variants The situation n players play a game. The rules are simple. The players enter a room blindfolded. A hat is put on each player’s head. The hat is either red or blue. Janne Peltola T-79.4001: The Hat Problem
  5. 5. Outline The Problem The Solution Variants The situation n players play a game. The rules are simple. The players enter a room blindfolded. A hat is put on each player’s head. The hat is either red or blue. The players must guess the color of their hat or pass. Janne Peltola T-79.4001: The Hat Problem
  6. 6. Outline The Problem The Solution Variants The situation n players play a game. The rules are simple. The players enter a room blindfolded. A hat is put on each player’s head. The hat is either red or blue. The players must guess the color of their hat or pass. The players may not communicate with each other. Janne Peltola T-79.4001: The Hat Problem
  7. 7. Outline The Problem The Solution Variants The situation n players play a game. The rules are simple. The players enter a room blindfolded. A hat is put on each player’s head. The hat is either red or blue. The players must guess the color of their hat or pass. The players may not communicate with each other. The players win if each non-passing guess is correct. Janne Peltola T-79.4001: The Hat Problem
  8. 8. Outline The Problem The Solution Variants The situation n players play a game. The rules are simple. The players enter a room blindfolded. A hat is put on each player’s head. The hat is either red or blue. The players must guess the color of their hat or pass. The players may not communicate with each other. The players win if each non-passing guess is correct. We wish to find a strategy which maximizes the probability p that the players win. Janne Peltola T-79.4001: The Hat Problem
  9. 9. Outline The Problem The Solution Variants Some history The problem was first proposed by T. Ebert (UCLA) in his Ph.D. thesis in 1998. There’s certain charm as a puzzle but the problem doesn’t seem terribly relevant. Janne Peltola T-79.4001: The Hat Problem
  10. 10. Outline The Problem The Solution Variants Some history The problem was first proposed by T. Ebert (UCLA) in his Ph.D. thesis in 1998. There’s certain charm as a puzzle but the problem doesn’t seem terribly relevant. However, there are connections between its efficient algorithmic solutions for n = 2k − 1 and coding theory. Janne Peltola T-79.4001: The Hat Problem
  11. 11. Outline The Problem The Solution Variants Some history The problem was first proposed by T. Ebert (UCLA) in his Ph.D. thesis in 1998. There’s certain charm as a puzzle but the problem doesn’t seem terribly relevant. However, there are connections between its efficient algorithmic solutions for n = 2k − 1 and coding theory. Thus far the problem has eluded tractable solutions for n = 2k − 1. Janne Peltola T-79.4001: The Hat Problem
  12. 12. Outline The Problem The Solution Variants 3 Players: a na¨ strategy ıve Suppose n = 3. The players are Alice, Bob and Chuck. Janne Peltola T-79.4001: The Hat Problem
  13. 13. Outline The Problem The Solution Variants 3 Players: a na¨ strategy ıve Suppose n = 3. The players are Alice, Bob and Chuck. If Alice and Bob pass, Chuck guesses correctly 50% of the time. Janne Peltola T-79.4001: The Hat Problem
  14. 14. Outline The Problem The Solution Variants 3 Players: a na¨ strategy ıve Suppose n = 3. The players are Alice, Bob and Chuck. If Alice and Bob pass, Chuck guesses correctly 50% of the time. Can we do better than this? Janne Peltola T-79.4001: The Hat Problem
  15. 15. Outline The Problem The Solution Variants 3 Players: a better strategy Each player looks at the other players’ R R R hats. R R B B B B B R R B B R R B B B R B R B R Janne Peltola T-79.4001: The Hat Problem
  16. 16. Outline The Problem The Solution Variants 3 Players: a better strategy Each player looks at the other players’ R R R hats. R R B If the hats are different, he/she passes. B B B B R R B B R R B B B R B R B R Janne Peltola T-79.4001: The Hat Problem
  17. 17. Outline The Problem The Solution Variants 3 Players: a better strategy Each player looks at the other players’ R R R hats. R R B If the hats are different, he/she passes. B B B If the hats are of the same color, he/she B R R picks the opposite one. B B R R B B B R B R B R Janne Peltola T-79.4001: The Hat Problem
  18. 18. Outline The Problem The Solution Variants 3 Players: a better strategy Each player looks at the other players’ R R R hats. R R B If the hats are different, he/she passes. B B B If the hats are of the same color, he/she B R R picks the opposite one. B B R R B B This results in p = 3 , a 50% increase. 4 B R B R B R Janne Peltola T-79.4001: The Hat Problem
  19. 19. Outline The Problem The Solution Variants 3 Players: a better strategy Each player looks at the other players’ R R R hats. R R B If the hats are different, he/she passes. B B B If the hats are of the same color, he/she B R R picks the opposite one. B B R R B B This results in p = 3 , a 50% increase. 4 B R B This strategy can be generalized for all R B R n = 2k − 1. Janne Peltola T-79.4001: The Hat Problem
  20. 20. Outline The Problem The Solution Variants 7 Players The size of the state space is 27 = 128. Janne Peltola T-79.4001: The Hat Problem
  21. 21. Outline The Problem The Solution Variants 7 Players The size of the state space is 27 = 128. The na¨ strategy can be applied. ıve Janne Peltola T-79.4001: The Hat Problem
  22. 22. Outline The Problem The Solution Variants 7 Players The size of the state space is 27 = 128. The na¨ strategy can be applied. ıve 63 However, there is a complex strategy to achieve p = 64 . Janne Peltola T-79.4001: The Hat Problem
  23. 23. Outline The Problem The Solution Variants 7 Players The size of the state space is 27 = 128. The na¨ strategy can be applied. ıve 63 However, there is a complex strategy to achieve p = 64 . Next, the general strategy will be defined. Janne Peltola T-79.4001: The Hat Problem
  24. 24. Outline The Problem The Solution Variants The Idea We have encoded certain states in C . We present a strategy to obtain winning sequences via C . We show that the strategy fails with probability p = 2−r . Janne Peltola T-79.4001: The Hat Problem
  25. 25. Outline The Problem The Solution Variants Definitions Consider a game of n = 2r − 1 players and 2 colors. Janne Peltola T-79.4001: The Hat Problem
  26. 26. Outline The Problem The Solution Variants Definitions Consider a game of n = 2r − 1 players and 2 colors. Let v = (1110100)T ∈ Zn be the state vector. 2 Definition Let H be an r × n matrix with entries in Z2 . Suppose that all possible nonzero columns occur exactly once. Then H is the check matrix (ie. Hv T = 0) of a code C . The code is called an [n, n − r ] Hamming code. Janne Peltola T-79.4001: The Hat Problem
  27. 27. Outline The Problem The Solution Variants The Strategy Recall the state vector v = (1110100)T ∈ Z7 2 Janne Peltola T-79.4001: The Hat Problem
  28. 28. Outline The Problem The Solution Variants The Strategy Recall the state vector v = (1110100)T ∈ Z7 2 1 1 1 1 0 0 0 We choose (for this case) H = 1 1 0 0 1 1 0 1 0 1 0 1 0 1 Janne Peltola T-79.4001: The Hat Problem
  29. 29. Outline The Problem The Solution Variants The Strategy Recall the state vector v = (1110100)T ∈ Z7 2 1 1 1 1 0 0 0 We choose (for this case) H = 1 1 0 0 1 1 0 1 0 1 0 1 0 1 For person 3, there are two possible v ’s: v0 = (1100100)T or v1 = (1110100)T . Janne Peltola T-79.4001: The Hat Problem
  30. 30. Outline The Problem The Solution Variants The Strategy Recall the state vector v = (1110100)T ∈ Z7 2 1 1 1 1 0 0 0 We choose (for this case) H = 1 1 0 0 1 1 0 1 0 1 0 1 0 1 For person 3, there are two possible v ’s: v0 = (1100100)T or v1 = (1110100)T . Person 3 uses the following rules to determine his call: Janne Peltola T-79.4001: The Hat Problem
  31. 31. Outline The Problem The Solution Variants The Strategy Recall the state vector v = (1110100)T ∈ Z7 2 1 1 1 1 0 0 0 We choose (for this case) H = 1 1 0 0 1 1 0 1 0 1 0 1 0 1 For person 3, there are two possible v ’s: v0 = (1100100)T or v1 = (1110100)T . Person 3 uses the following rules to determine his call: T T If Hv0 = 0 and Hv1 = 0, he passes Janne Peltola T-79.4001: The Hat Problem
  32. 32. Outline The Problem The Solution Variants The Strategy Recall the state vector v = (1110100)T ∈ Z7 2 1 1 1 1 0 0 0 We choose (for this case) H = 1 1 0 0 1 1 0 1 0 1 0 1 0 1 For person 3, there are two possible v ’s: v0 = (1100100)T or v1 = (1110100)T . Person 3 uses the following rules to determine his call: T T If Hv0 = 0 and Hv1 = 0, he passes T T If Hv0 = 0 and Hv1 = 0, he calls Red Janne Peltola T-79.4001: The Hat Problem
  33. 33. Outline The Problem The Solution Variants The Strategy Recall the state vector v = (1110100)T ∈ Z7 2 1 1 1 1 0 0 0 We choose (for this case) H = 1 1 0 0 1 1 0 1 0 1 0 1 0 1 For person 3, there are two possible v ’s: v0 = (1100100)T or v1 = (1110100)T . Person 3 uses the following rules to determine his call: T T If Hv0 = 0 and Hv1 = 0, he passes T T If Hv0 = 0 and Hv1 = 0, he calls Red T T If Hv0 = 0 and Hv1 = 0, he calls Blue Janne Peltola T-79.4001: The Hat Problem
  34. 34. Outline The Problem The Solution Variants Why is this a good idea? Lemma There is a unique codeword c so that dH (v , c) = 1. Proof. Existence: Let s = Hv T = 0. s is a column of H, therefore s = HeiT . H(v + ei )T = s + s = 0, so dH (v , c) = 1. Janne Peltola T-79.4001: The Hat Problem
  35. 35. Outline The Problem The Solution Variants Why is this a good idea? Lemma There is a unique codeword c so that dH (v , c) = 1. Proof. Uniqueness: Let c1 , c2 ∈ C so that dH (v , c1 ) = dH (v , c2 ) = 1. Therefore v + c1 = ei and v + c2 = ej for some (i, j). We have ei + ej = c1 + c2 ∈ C , so H(ei + ej )T = 0 HeiT = HejT so columns i and j must be equal, but there are no equal columns by definition ⇒ contradiction. Janne Peltola T-79.4001: The Hat Problem
  36. 36. Outline The Problem The Solution Variants Why is this NOT a good idea? Theorem The strategy is successful whenever Hv T = 0, but fails when Hv T = 0. Proof. We previously showed that there is a unique codeword c so that dH (v , c) = 1. Janne Peltola T-79.4001: The Hat Problem
  37. 37. Outline The Problem The Solution Variants Why is this NOT a good idea? Theorem The strategy is successful whenever Hv T = 0, but fails when Hv T = 0. Proof. We previously showed that there is a unique codeword c so that dH (v , c) = 1. If we iterate through the vectors vi = v + ei , we find that HviT = 0 only for one value of i. Janne Peltola T-79.4001: The Hat Problem
  38. 38. Outline The Problem The Solution Variants Why is this NOT a good idea? Theorem The strategy is successful whenever Hv T = 0, but fails when Hv T = 0. Proof. We previously showed that there is a unique codeword c so that dH (v , c) = 1. If we iterate through the vectors vi = v + ei , we find that HviT = 0 only for one value of i. Recall that we assumed that Hv T = 0, therefore the case T Hv0 = 0 precludes all other possibilities ⇒ v0 = v . Janne Peltola T-79.4001: The Hat Problem
  39. 39. Outline The Problem The Solution Variants Why is this NOT a good idea? Theorem The strategy is successful whenever Hv T = 0, but fails when Hv T = 0. Proof. We previously showed that there is a unique codeword c so that dH (v , c) = 1. If we iterate through the vectors vi = v + ei , we find that HviT = 0 only for one value of i. Recall that we assumed that Hv T = 0, therefore the case T Hv0 = 0 precludes all other possibilities ⇒ v0 = v . If Hv T = 0, no one would pass ⇒ the strategy fails. Janne Peltola T-79.4001: The Hat Problem
  40. 40. Outline The Problem The Solution Variants A helpful lemma Lemma An [n, k] linear code has 2k elements. Proof. An [n, k] linear code C has a check matrix H ∈ Zk×n . 2 Janne Peltola T-79.4001: The Hat Problem
  41. 41. Outline The Problem The Solution Variants A helpful lemma Lemma An [n, k] linear code has 2k elements. Proof. An [n, k] linear code C has a check matrix H ∈ Zk×n . 2 The elements of C are the columns + (000)T Janne Peltola T-79.4001: The Hat Problem
  42. 42. Outline The Problem The Solution Variants A helpful lemma Lemma An [n, k] linear code has 2k elements. Proof. An [n, k] linear code C has a check matrix H ∈ Zk×n . 2 The elements of C are the columns + (000)T By combinatorics, there are 2k elements. Janne Peltola T-79.4001: The Hat Problem
  43. 43. Outline The Problem The Solution Variants How often does one fail? Theorem The probability that players win by using the strategy is 1 − 2−r . Proof. There are 2n elements in Zn . 2 Janne Peltola T-79.4001: The Hat Problem
  44. 44. Outline The Problem The Solution Variants How often does one fail? Theorem The probability that players win by using the strategy is 1 − 2−r . Proof. There are 2n elements in Zn . 2 2k Therefore, the probability to find k specific elements is 2n . Janne Peltola T-79.4001: The Hat Problem
  45. 45. Outline The Problem The Solution Variants How often does one fail? Theorem The probability that players win by using the strategy is 1 − 2−r . Proof. There are 2n elements in Zn . 2 2k Therefore, the probability to find k specific elements is 2n . Therefore, the probability to hit an element v so that Hv T = 0 is 2−r . Janne Peltola T-79.4001: The Hat Problem
  46. 46. Outline The Problem The Solution Variants How often does one fail? Theorem The probability that players win by using the strategy is 1 − 2−r . Proof. There are 2n elements in Zn . 2 2k Therefore, the probability to find k specific elements is 2n . Therefore, the probability to hit an element v so that Hv T = 0 is 2−r . The complement case’s (Hv T = 0) probability is 1 − 2−r . Janne Peltola T-79.4001: The Hat Problem
  47. 47. Outline The Problem The Solution Variants Food for thought In the best tradition of the literature, I explore the opportunities for expanding the problem without providing actual solutions. One could consider k colors instead of 2. Janne Peltola T-79.4001: The Hat Problem
  48. 48. Outline The Problem The Solution Variants Food for thought In the best tradition of the literature, I explore the opportunities for expanding the problem without providing actual solutions. One could consider k colors instead of 2. One could consider n = 2r − 1. Janne Peltola T-79.4001: The Hat Problem
  49. 49. Outline The Problem The Solution Variants Food for thought In the best tradition of the literature, I explore the opportunities for expanding the problem without providing actual solutions. One could consider k colors instead of 2. One could consider n = 2r − 1. One could consider non-symmetrical color distributions. Janne Peltola T-79.4001: The Hat Problem
  50. 50. Outline The Problem The Solution Variants Food for thought In the best tradition of the literature, I explore the opportunities for expanding the problem without providing actual solutions. One could consider k colors instead of 2. One could consider n = 2r − 1. One could consider non-symmetrical color distributions. One could consider variants with more error tolerance (i.e. 1 error allowed). Could one solve n = 2r − 1 with relaxed conditions? Janne Peltola T-79.4001: The Hat Problem
  51. 51. Outline The Problem The Solution Variants Summary We found an interesting connection between coding theory and game theory. One can encode certain states and then make sure that there are only unique legal states which encode to nearby codewords. One can use this knowledge to develop a near-optimal strategy. Janne Peltola T-79.4001: The Hat Problem

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