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Game Theory and Games by pradeeban

VIEWS: 83 PAGES: 40

This is the presentation on Game Theory, applied to Games. This was presented by Xiao Chen, during the Distributed Systems class at KTH.

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									Xiao Chen

CHAPTER 6: GAMES
                  2013/11/14




WHAT IS A GAME?




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AN EXAM-OR-PRESENTATION GAME

 Goal: Decide whether to study for the exam, or to
  prepare for the presentation.
 Rule:

 1. For exam, if you study, your grade will be a 92,
  while if you don’t study, your grade will be a 80.
 2. For presentation, you’ll do it with a partner. If
  both of you prepare for the presentation, you’ll get
  a 100. If just one of you prepares, you’ll get a 92;
  and if neither of you prepares, you’ll get a 84.

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AN EXAM-OR-PRESENTATION GAME
 Both of you try to maximize your average grade.
 So…. What is the result?




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REASONING ABOUT BEHAVIOR IN THE GAME

 It reveals that no matter what your partner does,
  you should study for the exam.
 When a player has a strategy that is strictly better
  than all other options regardless of what the other
  player does, we will refer to it as a strictly
  dominant strategy.
 Since studying for the exam is also a strictly
  dominant strategy for both of you two, we should
  expect that the outcome will be for both of you to
  study, each getting an average grade of 88.

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BASIC INGREDIENTS OF A GAME.
 There is a set of participants, whom we call the
  players. (you and your partner)
 Each player has a set of options for how to
  behave; we will refer to these as the player’s
  possible strategies. (prepare presentation or
  study for the exam)
 For each choice of strategies, each player
  receives a payoff that can depend on the
  strategies selected by everyone. (average grade)

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THE PRISONER’S DILEMMA

 If you confess and your partner doesn’t confess, you
  will be released and your partner will be charged
  with the crime and sent to prison for 10 years.
 If you both confess, then we don’t need either of you
  to testify against the other, and you will both be
  convicted of the robbery. (4 years only)
 Finally, if neither of you confesses, we will charge
  each of you with resisting arrest for 1 year.
 “Do you want to confess?”

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THE PRISONER’S DILEMMA
   To make it clear….




 As a result, we should expect both suspects to
  confess, each getting a payoff of −4.
 What if the suspects could threaten each other?


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THE PRISONER’S DILEMMA

   Another example:




   What will you do if you are an athlete here?

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    THE PRISONER’S DILEMMA
   Exam-or-Presentation Game with an easier exam:
   We make the exam much easier, so that you’ll get a 100 on it if
    you study, and a 96 if you don’t.




   The downsides of the previous scenario no longer appear: the
    Prisoner’s Dilemma only manifests itself when the conditions
    are right.

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BEST RESPONSES AND DOMINANT STRATEGIES

 Best response: it is the best choice of one
  player, given a belief about what the other
  player will do.
 Player 1 has a strategy S; Player 2 has a
  strategy T. (S, T) is the strategies pair.
 P1(S, T) to denote the payoff to Player 1; same
  for P2(S, T).
 S for Player 1 is a best response :
                 P1(S, T) ≥ P1(S’, T)

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BEST RESPONSES AND DOMINANT STRATEGIES

   Specially, if
                    P1(S, T) > P1(S’, T)
                    
 S for Player 1 is a strict best response.
 We say that a dominant strategy (S)for Player 1 is
  a strategy that is a best response to every strategy
  of Player 2.
 We say that a strictly dominant strategy (S)for
  Player 1 is a strategy that is a strict best response
  to every strategy of Player 2.

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NASH EQUILIBRIUM

   A Three-Client Game: Two firms with three
    clients: A, B, C. Each firm has three possible
    strategies: whether to approach A, B, or C.




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NASH EQUILIBRIUM
 We say that this pair of strategies (S, T) is a
  Nash equilibrium if S is a best response to T,
  and T is a best response to S.
 The idea is that if the players choose strategies
  that are best responses to each other, then no
  player has an incentive to deviate to an
  alternative strategy — so the system is in a kind
  of equilibrium state, with no force pushing it
  toward a different outcome.

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BACK TO THE THREE-CLIENT GAME

 If Both Firms chooses A, then we can check
  that Firm 1 is playing a best response to Firm
  2’s strategy. So is Firm 2
 Hence, the pair of strategies (A,A) forms a Nash
  equilibrium. Moreover, we can check that this is
  the only Nash equilibrium.




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MULTIPLE EQUILIBRIA: COORDINATION GAMES

   A Coordination Game:




   Two Nash equilibria: (Power-Point, PowerPoint)
    and ( Keynote, Keynote)

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MULTIPLE EQUILIBRIA: COORDINATION GAMES

   Variants on the Basic Coordination Game




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MULTIPLE EQUILIBRIA: COORDINATION GAMES

   the Stag Hunt Game




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MULTIPLE EQUILIBRIA: COORDINATION GAMES

   Stag Hunt version Exam-or-Presentation Game




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MULTIPLE EQUILIBRIA: THE HAWK-DOVE GAME

   Players will also engage in a kind of “anti-
    coordination” activity.




   This game has two Nash equilibria: (D,H) and (H,D).
   So in equilibrium, we can expect that one will be
    aggressive and one will be passive, but we can’t
    predict who will follow which strategy.
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MULTIPLE EQUILIBRIA: THE HAWK-DOVE GAME

   Let’s see the Hawk-Dove version Exam-or-
    Presentation game.




   We cannot predict from the structure of the
    game alone who will play this passive role.
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MULTIPLE EQUILIBRIA: THE HAWK-DOVE GAME

   It is also referred to as the game of Chicken, to
    evoke the image of two teenagers racing their
    cars toward each other, daring each other to be
    the one to swerve out of the way.




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MIXED STRATEGIES

   Matching Pennies: two Players each hold a
    penny, and choose heads (H) or tails (T). Player
    1 loses his penny to player 2 if they match, and
    wins player 2’s penny if they don’t match.




                      Zero-Sum!

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MIXED STRATEGIES
 What do we learn from Matching Pennies?
 There is no pair of strategies that are best
  responses to each other. Also no Nash equilibrium
  for this game.
 How are games of this type played in real life?
 Simple to make opponents hard to predict what
  they will play.
 So we shouldn’t treat the strategies as simply H or
  T, but as ways of randomizing one’s behavior
  between H and T.

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MIXED STRATEGIES
 Model: Assume Player 1 plays H with probability p,
  and T with (1 − p). Same is q for Player 2.
 Payoff: for example, if Player 1 chooses the pure
  strategy H (p = 1) while Player 2 chooses a
  probability of q. then the expected payoff to Player
  1 is (−1)(q) + (1)(1 − q) = 1 − 2q.
 Similarly, if Player 1 chooses the pure strategy T
  while Player 2 chooses a probability of q, then the
  expected payoff to Player 1 is
 (1)(q) + (−1)(1 − q) = 2q − 1.

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MIXED STRATEGIES
 Payoffs are from the four pure outcomes (H,H), (H,
  T), (T,H), and (T, T).
 Equilibrium with Mixed Strategies:
 here is the key point, normally we can get 1 − 2q ≠
  2q − 1, then one of the pure strategies H or T is in
  fact the unique best response by Player 1 to a play
  of q by Player 2.
 What if 1 − 2q = 2q − 1 ? => q=0.5
 Thus, the pair of strategies p = 0.5 and q = 0.5 is
  the only possibility for a Nash equilibrium.

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MIXED STRATEGIES

 Interpreting the Mixed-Strategy Equilibrium for
  Matching Pennies:
 Players may be actively randomizing their
  actions
 The mixed strategies are better viewed as
  proportions within a population.
 The choice of probabilities is self-reinforcing (it
  is in equilibrium) across the entire population.

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MIXED STRATEGIES: EXAMPLES AND EMPIRICAL
ANALYSIS
 The Run-Pass Game:
 If the defense correctly matches the offense’s
  play, then the offense gains 0 yards.
 If the offense runs while the defense defends
  against the pass, the offense gains 5 yards.
 If the offense passes while the defense
  defends against the run, the offense gains 10
  yards.

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MIXED STRATEGIES: EXAMPLES AND EMPIRICAL
ANALYSIS
   Here we have:




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MIXED STRATEGIES: EXAMPLES AND EMPIRICAL
ANALYSIS
Now we are using probabilities:
 First, suppose the defense chooses a probability of q for
  defending against the pass. Then the expected payoff to
  the offense from passing is
              (0)(q) + (10)(1 − q) = 10 − 10q,
 while the expected payoff to the offense from running is
                  (5)(q) + (0)(1 − q) = 5q.
 To make the offense indifferent between its two
  strategies, we need to set 10−10q = 5q,
 and hence q = 2/3.


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MIXED STRATEGIES: EXAMPLES AND EMPIRICAL
ANALYSIS
 Next, suppose the offense chooses a probability of
  p for passing. Then the expected payoff to the
  defense from defending against the pass is
            (0)(p) + (−5)(1 − p) = 5p − 5,
 with the expected payoff to the defense from
  defending against the run is
            (−10)(p) + (0)(1 − p) = −10p.
 To make the defense indifferent between its two
  strategies, we need to set 5p−5 = −10p,
 and hence p = 1/3.


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MIXED STRATEGIES: EXAMPLES AND EMPIRICAL
ANALYSIS
   Thus, the only possible probability values that
    can appear in a mixed-strategy equilibrium are
    p = 1/3 for the offense, and q = 2/3 for the
    defense, and this in fact forms an equilibrium




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PARETO-OPTIMALITY AND SOCIAL OPTIMALITY

   Let’s look back to the first example…




 It will be interesting to classify outcomes in a game
  not just by their strategic or equilibrium properties,
 So are they “good for society”?


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PARETO-OPTIMALITY

 “A choice of strategies — one by each player — is
  Pareto-optimal if there is no other choice of
  strategies in which all players receive payoffs at
  least as high, and at least one player receives a
  strictly higher payoff.”
 Exam-or-Presentation Game: It shows that even if
  you and your partner realize there is a superior
  solution, there is no way to maintain it without a
  binding agreement between the two of you.

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SOCIAL OPTIMALITY
 “A choice of strategies — one by each player —
  is a social welfare maximizer (or socially
  optimal) if it maximizes the sum of the players’
  payoffs.”
 Outcomes that are socially optimal must also
  be Pareto-optimal.
 For example, the Exam-or-Presentation Game
  has three outcomes that are Pareto-optimal,
  but only one of these is the social optimum.

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CONCLUSION
 What is a Game?
 Reasoning about Behavior in a Game
 Best Responses and Dominant Strategies
 Nash Equilibrium
 Multiple Equilibria: Coordination Games
 Multiple Equilibria: The Hawk-Dove Game
 Mixed Strategies
 Mixed Strategies: Examples and Empirical Analysis
 Pareto-Optimality and Social Optimality



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JUST FOR FUN: THE PIRATE GAME




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THE PIRATE GAME
   5 pirates: A, B, C, D, E. 100 gold coins.
   Goal: how to distribute them?
   The pirates seniority order: A > B > C > D > E.
   Rules: The biggest guy proposes a distribution of coins.
    All the pirates. If approved, then distributed coins. If
    not, the proposer will be killed and next biggest pirate
    proposes and the game restarts.
   Each pirate wants to survive and they are very greedy.
    They also tend to kill the biggest guy who makes the
    deal if tie. They do not trust each other.
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THANK YOU VERY MUCH!
QUESTIONS?

								
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