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1. Free AI Kit
Game Theory
Fariz Darari, Ph.D.
doc.v04

2. What is Game Theory?
• Game theory is the study of the ways in which interacting choices of
economic agents produce outcomes with respect to the preferences
(or utilities) of those agents. (plato.stanford.edu)
• Game theory is a theoretical framework for conceiving social
situations among competing players. In some respects, game theory
is the science of strategy, or at least the optimal decision-making of
independent and competing actors in a strategic setting.
(investopedia.com)
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3. So, Game Theory is Basically ...
... how to make decisions in a multi-agent system.
Game Theory is leveraged to analyze agent decisions and measure the
expected utility for every decision, with the assumption that other
agents behave optimally.
As opposed to the turn-taking, fully observable adversarial search,
Game Theory works in a partially observable environment using
simultaneous moves.
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4. Two-Finger Morra Game
Evenia and Oddie, simultaneously, show one or two fingers.
Suppose that the sum of all the shown fingers is N.
Utilities:
• If N is even, then Evenia gets N dollar from Oddie.
• If N is odd, then Oddie gets N dollar from Evenia.
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5. Single-Move Games
• Agents make one-move decisions.
• Examples:
• Product pricing
• International relations
• Games in traditional sense: Two-finger Morra,
Rock-Paper-Scissors, etc
• Marriage? :)
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6. Components
• Players: Involved agents to make decisions. Can be two or
more.
• Actions: What agents can do. Action decisions need not be
unique among agents.
• Payoff Function: Utility values for every player for every
action combination by all players.
Such a payoff function can be represented by a matrix called:
Strategic form/normal form.
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7. Two-Finger Morra Game: Revisited
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Oddie: One Oddie: Two
Evenia: One E = 2, O = -2 E = -3, O = 3
Evenia: Two E = -3, O = 3 E = 4, O = -4
Evenia and Oddie, simultaneously, show one or two fingers.
Suppose that the sum of all the shown fingers is N.
Utilities:
• If N is even, then Evenia gets N dollar from Oddie.
• If N is odd, then Oddie gets N dollar from Evenia.

8. Game Strategies
• Every player must adopt and execute a strategy/policy.
• Pure strategy: Deterministic policy.
• Mixed strategy: Randomized policy, involves probability
distribution, [p : a, (1 – p): b]. For example, [0.4 : One, 0.6 : Two].
• A strategy profile is a strategy selection for every player with
some outcome value.
• A solution in Game Theory is a strategy profile where each
player takes a rational strategy.
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9. Prisoner's Dilemma
Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary
confinement with no means of communicating with the other. The prosecutors lack sufficient
evidence to convict the pair on the principal charge, but they have enough to convict both on
a lesser charge. Simultaneously, the prosecutors offer each prisoner a bargain. Each prisoner
is given the opportunity either to betray the other by testifying that the other committed the
crime, or to cooperate with the other by remaining silent. The possible outcomes are:
• If A and B each betrays the other, each of them serves two years in prison
• If A betrays B but B remains silent, A will be set free and B will serve three years in prison
• If A remains silent but B betrays A, A will serve three years in prison and B will be set free
• If A and B both remain silent, both of them will serve only one year in prison.
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10. Two members of a criminal gang are arrested and imprisoned. Each prisoner is in solitary
confinement with no means of communicating with the other. The prosecutors lack
sufficient evidence to convict the pair on the principal charge, but they have enough to
convict both on a lesser charge. Simultaneously, the prosecutors offer each prisoner a
bargain. Each prisoner is given the opportunity either to betray the other by testifying that
the other committed the crime, or to cooperate with the other by remaining silent. The
possible outcomes are:
• If A and B each betrays the other, each of them serves two years in prison
• If A betrays B but B remains silent, A will be set free and B will serve three years in prison
• If A remains silent but B betrays A, A will serve three years in prison and B will be set
free
• If A and B both remain silent, both of them will serve only one year in prison.
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11. Two members of a criminal gang are arrested and imprisoned. Each prisoner is in
solitary confinement with no means of communicating with the other. The
prosecutors lack sufficient evidence to convict the pair on the principal charge, but
they have enough to convict both on a lesser charge. Simultaneously, the
prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity
either to betray the other by testifying that the other committed the crime, or to
cooperate with the other by remaining silent. The possible outcomes are:
Strategy for B:
• If A stays silent: B stays silent (-1) vs. B bertrays (0)
• If A bertrays: B stays silent (-3) vs. B bertrays (-2)
Best strategy: B bertrays (always better than stays silent)
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12. Two members of a criminal gang are arrested and imprisoned. Each prisoner is in
solitary confinement with no means of communicating with the other. The
prosecutors lack sufficient evidence to convict the pair on the principal charge, but
they have enough to convict both on a lesser charge. Simultaneously, the
prosecutors offer each prisoner a bargain. Each prisoner is given the opportunity
either to betray the other by testifying that the other committed the crime, or to
cooperate with the other by remaining silent. The possible outcomes are:
Strategy for A?
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13. Dominant strategy for Prisoner's dilemma: To betray!*
*After all, they are criminals, so no code of conduct is required.
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14. Dominant Strategy
• Strategy s for player p strongly dominates strategy s0 if the outcome
of s is always better for p than the outcome of s0 no matter what is
done by the other players.
• Strategy s for player p weakly dominates strategy s0 if the outcome of
s is better for p than the outcome of s0 in at least one strategy
profile, and is no worse than s0 in the other strategy profiles.
• Rationality: Picking a dominant strategy.
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15. Exercise
What is each player's dominant strategy?
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*Player One's payoffs are in bold.

16. Exercise
What is each player's dominant strategy?
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*Player One's payoffs are in bold.
• Dominant strategy for Player One?
- If Player Two cooperates:
Player One Cooperate ($10) vs. Player One Cheat ($12)
- If Player Two cheats:
Player One Cooperate ($0) vs. Player One Cheat ($5)

17. Exercise
What is each player's dominant strategy?
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*Player One's payoffs are in bold.
• Dominant strategy for Player Two?
- If Player One cooperates:
Player Two Cooperate ($10) vs. Player Two Cheat ($12)
- If Player One cheats:
Player Two Cooperate ($0) vs. Player Two Cheat ($5)

18. Exercise
What is each player's dominant strategy?
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Oddie: One Oddie: Two
Evenia: One E = 2, O = -2 E = -3, O = 3
Evenia: Two E = -3, O = 3 E = 4, O = -4

19. Exercise
What is each player's dominant strategy?
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Oddie: One Oddie: Two
Evenia: One E = 2, O = -2 E = -3, O = 3
Evenia: Two E = -3, O = 3 E = 4, O = -4
Neither player has a dominant strategy.

20. Exercise
What is each player's dominant strategy?
20

21. Exercise
What is each player's dominant strategy?
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Neither player has a dominant strategy.
• If Player Two chooses rock, Player One should play paper
• If Player Two chooses paper, Player One responds with scissors
• If Player Two chooses scissors, Player One chooses rock

22. If every player has a dominant strategy, then the strategy combination
is called Dominant Strategy Equilibrium. 22
Dominant Strategy Equilibrium

23. If every player has a dominant strategy, then the strategy combination
is called Dominant Strategy Equilibrium. 23
Dominant Strategy Equilibrium
Dominant Strategy Equilibrium:
(Bertrays, Bertrays)

24. Nash Equilibrium
• Equilibrium is a concept such that no player would
gain more for changing her strategy if other players
do not change their strategy.
• The Nash Equilibrium is a decision-making theorem within Game Theory
that states a player can achieve the desired outcome by not deviating
from their initial strategy.
Note: Any dominant strategy equilibrium is always a Nash equilibrium!
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25. Pure Strategy Nash Equilibrium
1. Identify each player's optimal strategy in response to what other
players might do.
2. A Nash equilibrium is defined when all players are playing their
optimal strategies simultaneously.
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The dominant strategy equilibrium here is:
(Bertrays, Bertrays)
^And this is also the Nash equilibrium for the given problem.
Note: Any dominant strategy equilibrium is a Nash equilibrium.

26. Exercise
What is the Nash Equilibrium?
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27. Dilemma in Prisoner's Dilemma
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Nash Equilibrium: (Bertrays, Bertrays)
• The equilibrium outcome is worse than the outcome when both
players stay silent.
• An outcome is Pareto-optimal if there is no other outcome wanted by
all players.
• An outcome o is Pareto-dominated by outcome o' if every player
chooses o'. For example, (-2, -2) is Pareto-dominated by (-1, -1).

28. Putting all together
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• Is there a dominant strategy?
• Nash Equilibrium?
• Which is best?
Best: One Best: Two
Acme: One A = 9, B = 9 A = -3, B = -1
Acme: Two A = -4, B = -1 A = 5, B = 5

29. Putting all together
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• Is there a dominant strategy? No!
• Nash Equilibrium? Two equilibria, that is, (One, One) and (Two, Two)
• Which is best? Pareto-optimal Nash Equilibrium = (One, One)
Best: One Best: Two
Acme: One A = 9, B = 9 A = -3, B = -1
Acme: Two A = -4, B = -1 A = 5, B = 5

30. Mixed Strategy
• Some problems do not have any pure strategy Nash equilibrium.
Recall: Two-finger Morra Game
• A mixed strategy could come into handy: Adding probabilities to our decisions.
• For example, Evenia could pick One with probability p, and Two with probability
(1 - p).
• Goal of a mixed strategy: To reach an equilibrium such that the opponent of the
player cannot make a deteministic choice to benefit from any of the player's
strategies. In other words, the player's strategies are indifferent. 30
Oddie: One Oddie: Two
Evenia: One E = 2, O = -2 E = -3, O = 3
Evenia: Two E = -3, O = 3 E = 4, O = -4

31. Mixed Strategy Equilibrium
• Suppose Oddie picks One with probability p and Two with probability (1 - p).
• If Evenia picks One, then Evenia would get expected payoff: 2p - 3(1 - p) = 5p - 3
• If Evenia picks Two, then Evenia would get expected payoff: -3p + 4(1 - p) = 4 - 7p
• Mixed Strategy Equilibrium: Strategy profile (of probabilities) that is best for two players
• The equilibrium can be achieved when all the choices are indifferent to the opponent.
So, in this case, it is solving: 5p – 3 = 4 – 7p. We then have that: p = 7/12.
• Proof: Suppose that Oddie picks One with probability 7/12. We would like to know the expected
payoffs for Evenia.
• If Evenia picks One, then the expected payoff = 5(7/12) – 3 = -(1/12)
• If Evenia picks Two, then the expected payoff = 4 – 7(7/12) = -(1/12)
So, whatever strategy is chosen by Evenia, there is no difference.
Exercise: What happens if Oddie picks One with probability 3/4?
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Oddie: One Oddie: Two
Evenia: One E = 2, O = -2 E = -3, O = 3
Evenia: Two E = -3, O = 3 E = 4, O = -4

32. Mixed Strategy Equilibrium (cont.)
• Suppose Evenia picks One with probability p and Two with probability (1 - p). What would be the
best strategy for Evenia?
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Oddie: One Oddie: Two
Evenia: One E = 2, O = -2 E = -3, O = 3
Evenia: Two E = -3, O = 3 E = 4, O = -4

33. Mixed Strategy Equilibrium (cont.)
• Suppose Evenia picks One with probability p and Two with probability (1 - p). What would be the
best strategy for Evenia? It's the same as Oddie (due to the symmetry of the payoff matrix),
picking p = 7/12.
• In this case, Oddie's best strategy is p = 7/12, and Evenia's one is also p = 7/12.
• The equilibrium for mixed strategy cases is called: Maximin equilibrium.
• Therefore, the maximin equilibrium for the above payoff matrix is: (7/12, 7/12)
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Oddie: One Oddie: Two
Evenia: One E = 2, O = -2 E = -3, O = 3
Evenia: Two E = -3, O = 3 E = 4, O = -4

34. Credits
• Fasilkom UI AI Slides from previous years
• Wikipedia
• Investopedia
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35. 35Thanks!