GAMING WITH MATHEMATICS E = mc²
AGENDA <ul><li>Game Theory </li></ul><ul><li>Game 1: Monty Hall </li></ul><ul><li>Game 2: Pick Up Sticks </li></ul><ul><li...
GAME THEORY <ul><li>A branch of applied mathematics used in social sciences, economics, biology, computer science and phil...
<ul><li>Traditional applications attempt to find sets of strategies where individuals are unlikely to change their behavio...
APPLICATIONS OF GAME THEORY <ul><li>Political Science </li></ul><ul><li>Economics and Business </li></ul><ul><li>Biology <...
GAME 1: MONTY HALL <ul><li>Player is given a choice of three doors where  only one  contains a prize </li></ul><ul><li>The...
<ul><li>Player is given a second chance if he or she would like to switch to the other un-picked door </li></ul><ul><li>Th...
<ul><li>http://www.shodor.org/interactivate/activities/SimpleMontyHall/?version=1.6.0_03&browser=MSIE&vendor=Sun_Microsyst...
<ul><li>You think the chance of winning is 50-50? </li></ul><ul><li>Think again… </li></ul>
<ul><li>The probability of winning by using the switching technique is in fact 2/3 </li></ul><ul><li>While the odds for wi...
<ul><li>The probability of picking the wrong door in the initial stage of the game is 2/3 </li></ul><ul><li>If the player ...
GAME 2: PICK UP STICKS <ul><li>Suppose that there are 10 sticks on the board </li></ul><ul><li>Player 1 and 2 will take al...
LET’S HAVE A TRIAL PLAY! <ul><li>Any 2 volunteers? </li></ul>
STRATEGY TO WIN <ul><li>At any stage, after player 1's move, the number of sticks left on board must be a multiple of 3 </...
WHAT DOES SUCH A STRATEGY ENSURE?  <ul><li>Consider the second last round, player 1 would have played so that 3 sticks are...
GAME TREE
IN GENERAL… <ul><li>In a two-player  Pick Up Stick  game, each of the two players can pick 1 through K sticks, with M stic...
FIBONACCI SERIES <ul><li>a sequence of numbers named after Leonardo of Pisa, known as Fibonacci in 1202 </li></ul>
<ul><li>The sequence is defined by the following recurrence relation: </li></ul><ul><li>i.e. 0, 1, 1, 2, 3, 5, 8, 13, 21, ...
APPLICATIONS OF FIBONACCI SERIES <ul><li>Appear in biological settings </li></ul><ul><li>Numerous poorly substantiated cla...
FIBONACCI RABBIT <ul><li>Suppose a newly-born pair of rabbits, one male, one female, are put in a field </li></ul><ul><li>...
<ul><li>At the end of the first month, they mate, but there is still one only 1 pair.  </li></ul><ul><li>At the end of the...
GAME 3: BRICK WALL <ul><li>Build a brick wall, using bricks with a length twice as long as its height </li></ul><ul><li>Th...
<ul><li>Let f(n) be the number of patterns for walls of height 2 and length n </li></ul><ul><li>The solutions we have at p...
<ul><li>The number of patterns of length n is the sum of the number of patterns of length (n-1) and the number of patterns...
GAME 4: CENTIPEDE GAME <ul><li>One of the games used in Game Theory </li></ul><ul><li>The game was developed by Rosenthal ...
RULES <ul><li>Two players, A and B </li></ul><ul><li>Players do not cooperate </li></ul><ul><li>Initially A has two piles ...
<ul><li>Each player has two options of moves: either take the larger pile and give the smaller pile or pass both piles acr...
APPLICATIONS <ul><li>Aimed to understand legislative bargaining concepts in political science </li></ul><ul><li>Laboratory...
LET’S PLAY! <ul><li>Any two volunteers? </li></ul><ul><li>Condition: Maximum number of moves is 10 </li></ul>
STRATEGY <ul><li>Decision to move  </li></ul><ul><li>right is known as  </li></ul><ul><li>a &quot;pass“ </li></ul><ul><li>...
MATHEMATICAL REASONING <ul><li>If B passes, he or she will get 16 while A gets 64. By &quot;taking“, B gets 32 while A onl...
<ul><li>Such reasoning is known as Nash Equilibrium (backward induction) </li></ul><ul><li>As the stakes increase, the pla...
<ul><li>By continuing playing, the winnings increases </li></ul><ul><li>But the risk is that your opponent might get more ...
NASH EQUILIBRIUM <ul><li>If there is a set of strategies with the property that no player can benefit by changing her stra...
EXAMPLE <ul><li>Two suspected persons, A and B, are arrested by the police. Both of them are interrogated separately. </li...
<ul><li>If they betray each other, both will receive a 5-year arrest </li></ul><ul><li>Each prisoner must make the choice ...
<ul><li>Payoff matrix of prisoner dilemma </li></ul>
<ul><li>Each prisoner wants to minimize the time of imprisonment </li></ul><ul><li>Betraying is the dominant strategy </li...
<ul><li>If you knew the other prisoner would stay silent, your best move is to betray </li></ul><ul><li>If you knew the ot...
BACK TO NASH EQUILIBRIUM <ul><li>Condition 1: The player who did not change has no better strategy in the new circumstance...
<ul><li>If both cases are met, the equilibrium is said to be stable </li></ul><ul><li>If condition 1 does not hold, then t...
<ul><li>That’s the end of our presentation. </li></ul><ul><li>THANK YOU </li></ul>
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Game theory & Fibonacci series

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Game theory & Fibonacci series

  1. 1. GAMING WITH MATHEMATICS E = mc²
  2. 2. AGENDA <ul><li>Game Theory </li></ul><ul><li>Game 1: Monty Hall </li></ul><ul><li>Game 2: Pick Up Sticks </li></ul><ul><li>Fibonacci Series </li></ul><ul><li>Game 3: Brick Wall </li></ul><ul><li>Game 4: Centipede Game </li></ul><ul><li>Nash Equilibrium </li></ul>
  3. 3. GAME THEORY <ul><li>A branch of applied mathematics used in social sciences, economics, biology, computer science and philosophy </li></ul><ul><li>Attempts to mathematically capture behavior in strategic situations </li></ul>
  4. 4. <ul><li>Traditional applications attempt to find sets of strategies where individuals are unlikely to change their behavior </li></ul><ul><li>Also known as equilibria </li></ul><ul><li>These equilibrium concepts are motivated differently depending on the field of application </li></ul>
  5. 5. APPLICATIONS OF GAME THEORY <ul><li>Political Science </li></ul><ul><li>Economics and Business </li></ul><ul><li>Biology </li></ul><ul><li>Computer Science and Logic </li></ul><ul><li>Philosophy </li></ul>
  6. 6. GAME 1: MONTY HALL <ul><li>Player is given a choice of three doors where only one contains a prize </li></ul><ul><li>The player first chooses an initial door </li></ul><ul><li>The host will then reveal an empty door among the other two un-picked doors </li></ul>
  7. 7. <ul><li>Player is given a second chance if he or she would like to switch to the other un-picked door </li></ul><ul><li>The host will open up the final chosen door and reveal the result </li></ul><ul><li>The player wins if the door he or she picked in the end contains the prize </li></ul>
  8. 8. <ul><li>http://www.shodor.org/interactivate/activities/SimpleMontyHall/?version=1.6.0_03&browser=MSIE&vendor=Sun_Microsystems_Inc . </li></ul>
  9. 9. <ul><li>You think the chance of winning is 50-50? </li></ul><ul><li>Think again… </li></ul>
  10. 10. <ul><li>The probability of winning by using the switching technique is in fact 2/3 </li></ul><ul><li>While the odds for winning by not switching is 1/3 </li></ul><ul><li>Let’s take a closer look… </li></ul>
  11. 11. <ul><li>The probability of picking the wrong door in the initial stage of the game is 2/3 </li></ul><ul><li>If the player picks the wrong door initially, the host must reveal the remaining empty door in the second stage of the game, then the last door must be the prized one </li></ul><ul><li>Thus, if the player switches over after picking the wrong door initially, he or she will win the prize </li></ul>
  12. 12. GAME 2: PICK UP STICKS <ul><li>Suppose that there are 10 sticks on the board </li></ul><ul><li>Player 1 and 2 will take alternate turns to pick up the sticks </li></ul><ul><li>In each turn, the player may either pick up 1 or 2 sticks only </li></ul><ul><li>The player who picks the last stick wins. </li></ul>
  13. 13. LET’S HAVE A TRIAL PLAY! <ul><li>Any 2 volunteers? </li></ul>
  14. 14. STRATEGY TO WIN <ul><li>At any stage, after player 1's move, the number of sticks left on board must be a multiple of 3 </li></ul><ul><li>Such that at the beginning, player 1 will only picks up 1 stick (so 9 sticks are left) </li></ul><ul><li>Subsequently, if player 2 picks up 1 stick, player 1 will pick up 2 </li></ul><ul><li>If player 2 picks up 2 sticks, then player 1 picks up 1 </li></ul>
  15. 15. WHAT DOES SUCH A STRATEGY ENSURE? <ul><li>Consider the second last round, player 1 would have played so that 3 sticks are left </li></ul><ul><li>Hence, no matter how many stick/s player 2 picks, player 1 will always has the winning move </li></ul>
  16. 16. GAME TREE
  17. 17. IN GENERAL… <ul><li>In a two-player Pick Up Stick game, each of the two players can pick 1 through K sticks, with M sticks initially </li></ul><ul><li>A winning strategy corresponds to picking sticks such that the number of sticks left is a multiple of (K+1) </li></ul><ul><li>The player who can ensure this strategy will surely win the game. </li></ul><ul><li>In the game of Pick up Stick mentioned earlier on, we have M=10 and K=2. </li></ul>
  18. 18. FIBONACCI SERIES <ul><li>a sequence of numbers named after Leonardo of Pisa, known as Fibonacci in 1202 </li></ul>
  19. 19. <ul><li>The sequence is defined by the following recurrence relation: </li></ul><ul><li>i.e. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987 … </li></ul>
  20. 20. APPLICATIONS OF FIBONACCI SERIES <ul><li>Appear in biological settings </li></ul><ul><li>Numerous poorly substantiated claims of Fibonacci numbers are also found in popular sources </li></ul><ul><li>Let’s see one example… </li></ul>
  21. 21. FIBONACCI RABBIT <ul><li>Suppose a newly-born pair of rabbits, one male, one female, are put in a field </li></ul><ul><li>They are able to mate at the age of one month so that at the end of its second month a female can produce another pair of rabbits </li></ul><ul><li>Suppose that our rabbits never die and that the female always produces one new pair (one male, one female) every month from the second month onwards </li></ul>
  22. 22. <ul><li>At the end of the first month, they mate, but there is still one only 1 pair. </li></ul><ul><li>At the end of the second month the female produces a new pair, so now there are 2 pairs of rabbits in the field. </li></ul><ul><li>At the end of the third month, the original female produces a second pair, making 3 pairs in all in the field. </li></ul><ul><li>At the end of the fourth month, the original female has produced yet another new pair, the female born two months ago produces her first pair also, making 5 pairs. </li></ul>
  23. 23. GAME 3: BRICK WALL <ul><li>Build a brick wall, using bricks with a length twice as long as its height </li></ul><ul><li>The wall is to be two units tall </li></ul><ul><li>We can build the wall in a number of patterns, depending on how long we want it </li></ul>
  24. 24. <ul><li>Let f(n) be the number of patterns for walls of height 2 and length n </li></ul><ul><li>The solutions we have at present are </li></ul>f(1) = 1 f(2) = 2 f(n)=f(n-1)+f(n-2) . . .
  25. 25. <ul><li>The number of patterns of length n is the sum of the number of patterns of length (n-1) and the number of patterns of length (n-2) </li></ul><ul><li>In terms of the &quot;f&quot; notation in mathematics, we have f(n) = f(n-1) + f(n-2) </li></ul><ul><li>Since f(1) = 1 and f(2) = 2, f(n) numbers are indeed the Fibonacci numbers! </li></ul>
  26. 26. GAME 4: CENTIPEDE GAME <ul><li>One of the games used in Game Theory </li></ul><ul><li>The game was developed by Rosenthal in 1981 </li></ul>
  27. 27. RULES <ul><li>Two players, A and B </li></ul><ul><li>Players do not cooperate </li></ul><ul><li>Initially A has two piles of coins: one pile contains 4 coins and the other 1 coin </li></ul>
  28. 28. <ul><li>Each player has two options of moves: either take the larger pile and give the smaller pile or pass both piles across </li></ul><ul><li>Each time the pile of coins pass across the table, the number of coins in it doubles </li></ul><ul><li>The game continues for a fixed number of rounds or until a player decides to end the game by keeping the larger pile and giving the other away </li></ul>
  29. 29. APPLICATIONS <ul><li>Aimed to understand legislative bargaining concepts in political science </li></ul><ul><li>Laboratory tested theory would be more accurate and can be precise in predicting actions in an empirical way </li></ul><ul><li>Used to predict behaviors of political balances </li></ul>
  30. 30. LET’S PLAY! <ul><li>Any two volunteers? </li></ul><ul><li>Condition: Maximum number of moves is 10 </li></ul>
  31. 31. STRATEGY <ul><li>Decision to move </li></ul><ul><li>right is known as </li></ul><ul><li>a &quot;pass“ </li></ul><ul><li>Moving down is described as a &quot;take“ </li></ul><ul><li>The numbers at the end of the down arrow are the payoffs to A and B respectively </li></ul>
  32. 32. MATHEMATICAL REASONING <ul><li>If B passes, he or she will get 16 while A gets 64. By &quot;taking“, B gets 32 while A only gets 8 </li></ul><ul><li>At the prior vertex, A knowing what B will do, will chose to take first </li></ul><ul><li>Therefore, it is suggested that the first player take right at the start </li></ul>
  33. 33. <ul><li>Such reasoning is known as Nash Equilibrium (backward induction) </li></ul><ul><li>As the stakes increase, the play tends to approach Nash equilibrium </li></ul><ul><li>But it will never reach an equilibrium </li></ul>
  34. 34. <ul><li>By continuing playing, the winnings increases </li></ul><ul><li>But the risk is that your opponent might get more than you at the end of the game </li></ul>
  35. 35. NASH EQUILIBRIUM <ul><li>If there is a set of strategies with the property that no player can benefit by changing her strategy while the other players keep their strategies unchanged, then that set of strategies and the corresponding payoffs constitute the Nash Equilibrium </li></ul>
  36. 36. EXAMPLE <ul><li>Two suspected persons, A and B, are arrested by the police. Both of them are interrogated separately. </li></ul><ul><li>If one betrays the other and the other remains silent, the betrayer is released and the silent suspect gets penalized with the full 10 year imprisonment </li></ul><ul><li>If both remain silent, both suspects will be sentenced to 6 months in jail </li></ul>
  37. 37. <ul><li>If they betray each other, both will receive a 5-year arrest </li></ul><ul><li>Each prisoner must make the choice of whether to betray the other or to remain silent </li></ul><ul><li>However, neither of them knows what choice the other prisoner will make </li></ul><ul><li>So how should the prisoners act? </li></ul>
  38. 38. <ul><li>Payoff matrix of prisoner dilemma </li></ul>
  39. 39. <ul><li>Each prisoner wants to minimize the time of imprisonment </li></ul><ul><li>Betraying is the dominant strategy </li></ul>
  40. 40. <ul><li>If you knew the other prisoner would stay silent, your best move is to betray </li></ul><ul><li>If you knew the other prisoner would betray, your best move is still to betray </li></ul><ul><li>Thus, they will get a lower payoff than what they would get by staying silent </li></ul>
  41. 41. BACK TO NASH EQUILIBRIUM <ul><li>Condition 1: The player who did not change has no better strategy in the new circumstance </li></ul><ul><li>Condition 2: The player who changes is now playing with a strictly worse strategy </li></ul>
  42. 42. <ul><li>If both cases are met, the equilibrium is said to be stable </li></ul><ul><li>If condition 1 does not hold, then the equilibrium is unstable </li></ul><ul><li>If only condition 1 holds, then there are likely to be an infinite number of optimal strategies for the player who changed </li></ul>
  43. 43. <ul><li>That’s the end of our presentation. </li></ul><ul><li>THANK YOU </li></ul>

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