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A game theoretical approach Matteo Venanzi

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1 A game theoretical approach Matteo Venanzi

2  Utilities and preferences  Multiagent encounters ◦ Solution concepts  The Prisoner’s Dilemma ◦ IPD in a social network

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4  Modelling preferences: ◦ Outcomes (states of the world): Ω = {w 1, w 2,...} ◦ Utility function: u i : Ω → R  Self-interested agents: Each agent has its own preferences and desires over the states of the world (non-cooperative game theory).

5 Utility functions lead to preference orderings over outcomes Preference over w Strict preference over w Properties Reflexivity Transitivity Comparability

6  Interaction as a game: the states of the world can be seen as the outcomes of a game.  Assume we have just two agents (players) Ag={i,j},  The final outcome in Ω depends on the combination of actions selected by each agent.  State transformer function:

7  Normal-form game (or strategic-form game): A strategic interaction is familiarly represented in game theory as a tuple (N, A, u), where: ◦ N is (finite) the set of player. ◦ A = A 1 x A 2 x... x A n where A i is the set of actions available to player i ◦ U = (u 1, u 2,..., u n ) utility functions of each player  Payoff matrix: n-dimentional matrix with cells enclosing the values of the utility functions for each player i cooperatesi defects j cooperates1, 11, 4 j defects4,14,14, 4

8 Apparently: “just” maximize the expected payoff (single-agent point of view). – In most cases unfeasible because the individual best strategy depends on the choices of others (multi-agent point of view) In practice: answer in solution concepts: – Dominant strategies – Pareto optimality – Nash equilibrium  Rational agent’s strategy: Given a game scenario, how a rational agent will act?

9  Best response: given the player j’s strategy s j, the player i’s best response to s j is the strategy s i that gives the highest payoff for player i. i: Ci: D j: C1, 11, 4 j: D4,44,44, 1 Example: j plays C----->i’s best response = D j plays D----->i’s best response = C

10  Dominant strategy: A strategy s i * is dominant for player i if no matter what strategy s j agent j chooses, i will do at least as well playing s i * as it would doing anything else. (s i * is dominant if it is the best response to all of agent j’s strategies.) i: Ci: D j: C1, 41, 1 j: D4,14,14, 4 Example: - D is the dominant strategy for player j -There are not dominant strategies for player i Dominated strategy can be removed from the table

11  Dominant strategy: A strategy s i * is dominant for player i if no matter what strategy s j agent j chooses, i will do at least as well playing s i * as it would doing anything else. (s i * is dominant if it is the best response to all of agent j’s strategies.) i: Ci: D j: C1, 41, 1 j: D4,14,14, 4 Example: - D is the dominant strategy for player j -There are not dominant strategies for player i Dominated strategy can be removed from the table i: Ci: D j: C1, 41, 1 j: D4,14,14, 4

12  Dominant strategy: A strategy s i * is dominant for player i if no matter what strategy s j agent j chooses, i will do at least as well playing s i * as it would doing anything else. (s i * is dominant if it is the best response to all of agent j’s strategies.) Example: - D is the dominant strategy for player j -There are not dominant strategies for player i Dominated strategy can be removed from the table i: Ci: D j: D4,14,14, 4

13  Pareto optimality (or Pareto efficiency): An outcome for the game is said to be Pareto optimal (or Pareto efficient) if there is no other outcome that makes one agent better off without making another agent worse off. If an outcome w is not Pareto optimal, then there is another outcome w’ that makes everyone as happy, if not happier, than w. “Reasonable” agents would agree to move to w’ in this case. (Even if I don’t directly benefit from w’, you can benefit without me suffering.)

14  Pareto optimality (or Pareto efficiency): An outcome for the game is said to be Pareto optimal (or Pareto efficient) if there is no other outcome that makes one agent better off without making another agent worse off. If an outcome w is not Pareto optimal, then there is another outcome w’ that makes everyone as happy, if not happier, than w. “Reasonable” agents would agree to move to w’ in this case. (Even if I don’t directly benefit from w’, you can benefit without me suffering.) Girl: whole cakeGirl: half cakeGirl: nothing Boy: whole cake Boy: Half cake Boy: Nothing

15  Pareto optimality (or Pareto efficiency): An outcome for the game is said to be Pareto optimal (or Pareto efficient) if there is no other outcome that makes one agent better off without making another agent worse off. If an outcome w is not Pareto optimal, then there is another outcome w’ that makes everyone as happy, if not happier, than w. “Reasonable” agents would agree to move to w’ in this case. (Even if I don’t directly benefit from w’, you can benefit without me suffering.) Girl: whole cakeGirl: half cakeGirl: nothing Boy: whole cake Boy: Half cake Boy: Nothing Not Pareto efficient Pareto efficient Not Pareto efficient Pareto efficient Pareto optimality is a nice proberty but not very useful for selecting strategies

16 Nash equilibrium for pure strategies: Two strategies s1 and s2 are in Nash equilibrium if: 1.under the assumption that agent i plays s1, agent j can do no better than play s2; AND 2.under the assumption that agent j plays s2, agent i can do no better than play s1.  Neither agent has any incentive to deviate from a Nash equilibrium.  Nash equilibrium represents the “Rational” outcome of a game played by self-interested agents. Unfortunately: ◦ Not every interaction scenario has a Nash equilibrium. ◦ Some interaction scenarios have more than one Nash equilibrium. But: Nash’s Theorem (1951): Every game with a finite number of players and a finite set of possible strategies has at least one Nash equilibrium in mixed strategies. However: “The complexity of finding a Nash equilibrium is the most important concrete open question on the boundary of P today” [Papadimitriou, 2001] John Forbes Nash, Jr. Bluefield Nobel Prize for Economics, 1994 “A beautiful mind”

17  Battle of the sexes game: husband and wife wish to go to the movies and they are undecided between “lethal Weapon (LW)” and “Wondrous Love (WL)”. They much prefer to go together rather than separate to the movie, altought the wife prefers LW and the husband prefers (WL)”. Easy way to find pure-strategy Nash equilibria in a payoff matrix: Take the cell where the first number is the maximum of the column and check if the second number is the maximum of that row. husband: LW husband: WL wife: LW2, 10, 0 wife: WL0,00,01, 2

18 husband: LW husband: WL wife: LW2, 10, 0 wife: WL0,00,01, 2  Battle of the sexes game: husband and wife wish to go to the movies and they are undecided between “lethal Weapon (LW)” and “Wondrous Love (WL)”. They much prefer to go together rather than separate to the movie, altought the wife prefers LW and the husband prefers (WL)”. Easy way to find pure-strategy Nash equilibria in a payoff matrix: Take the cell where the first number is the maximum of the column and check if the second number is the maximum of that row.

19  Max-min strategy: the maxmin strategy for player i is the strategy s i that maximizes the i’s payoff in the worst case. (Minmax is the dual strategy) Example: (zero-sum game: the sum of the utilities is always zero) Player 2: APlayer 2: B Player 1: A2, -23, -3 Player 1: B0,00,04, -4 From the player 1’s point of view: - If he plays A, the min payoff is 2. - If he plays B, the min payoff is 0.  Player 1 chooses A. With the same reasoning:  player 2 chooses A too. The “maxmin solution” is (A, A). It is also the (only) Nash equilibrium of the game. In fact: Minmax Theorem ( von Neuman, 1928 ) : In any finite, two-players, zero-sum game maxmin strategies, minmax strategies and Nash equilibria coincide.

20 Two men are collectively charged with a crime and held in separate cells, with no way of meeting or communicating. They are told that: ◦ if one confesses and the other does not, the confessor will be freed, and the other will be jailed for five years; ◦ if both confess, then each will be jailed for two years. ◦ Both prisoners know that if neither confesses, then they will each be jailed for one year.  Well-known (and ubiquitous) puzzle in game theory.  The fundamental paradox of multi-agent interactions. Payoff matrix for prisoner’s dilemma: Player 2 confessesPlayer 2 does not confess Player 1 confesses1, 15, 0 Player 1 does not confess0,50,53, 33, 3

21 (Confession = defection, No confession = cooperation) Solution concepts: ◦ D is a dominant strategy. ◦ (D, D) is the only Nash equilibrium. ◦ All outcomes except (D;D) are Pareto optimal. Game theorist’s conclusions: ◦ In the worst case Defection guarantees a highest payoff so Defection is for both players the rational move to make.  Both agents defects and get payoff = 2 Why dilemma..? ◦ They both know they could make better off by both cooperating and each get payoff =3. Cooperation is too dangerous!!  In the one-shot prisoner’s dilemma the individual rational choice is defection Player 2 DefectsPlayer 2 Cooperates Player 1 Defects1, 15, 0 Player 1 Cooperates0,50,53, 33, 3

22  Observation: If the PD is repeated a number of time (so, I will meet my opponent again) then the incentive to Defect appears to decrease and Cooperation can finally emerge (Iterated Prisoner’s Dilemma).  Axelrod in 1985 organized two computer tournament, inviting participants to submit their strategies to play the IPD. ◦ The length of the game was unknown to the players. ◦ 15 strategies submitted. ◦ Best strategies:  SPITEFUL: Cooperate until receive a defection, then always defect  MISTRUST: Defect, then play the last opponent’s move.  TIT FOR TAT: Cooperate on 1st round, then play what the opponent played on the previous round. ◦ TIT_FOR_TAT won both the tournaments!!! (Still today, nobody was able to “clearly” defeat TFT)

23  Axelrod also suggests the following rules for a successful strategy for the IPD:  Don’t be envious: Remember the PD is not a zero-sum game. Don’t play as you would strike your opponent!  Be nice: Never be the first to defect.  Retaliate appropriately: There must be a proper measure in rewarding cooperation and punishing defection.  Don’t be too clever: The performance of a strategy are not necessarily related to its complexity. Even though TFT was the best strategy in the Axelrod’s tournament, it doesn’t mean it the Optimal Strategy for the IPD. In fact: Axelrod’s Theorem: If the game is long enough and the players do care about their future encounters, a universal optimal strategy for the Iterated Prisoner’s Dilemma does not exist.

24  Imagine agents playing the IPD, arranged as a graph where the edges represents rating relationships.  The agent can use the knowledge and experience of his neighbours for retrieving ratings about his opponent and inferring his reputation. Rating propagation: Mechanism to propagate ratings from one node to another within the network. ◦ possible solution: the propagated rating the product of the weights on the edges along the path connecting the two players. In case of Multiple Path, take the one with the highest weight product.  Strategy: Cooperate with probability P max

25 Random Network  TrustRate-Agents get all around the same score  The ranking keeps the features of the Axelrod’s Tournament

26 Dense Graph  Increases the number of ratings.  There is always a nice path between any couple of players.

27 Get new agents into the game in progress Agent 11 inserted at round 55/100  The network helps the new player to immediately recognize the opponents with who cooperate or to avoid.  Agent 11 scores the same points than the starting players.

28  Conclusions: ◦ Acquaintance’s network allows to promote cooperation in the IPD with a distributed approach.  No data overload on a single agent.  Information spread through the whole network. ◦ Incoming agents are immediately in conditions to get into the logics of cooperation.  If surrounded by some nice neighbourhood, the agent is able to immediately distinguish cooperative players from defective players, even without any prior experience.

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30  An introduction to Multiagent Systems (2 nd edition). M. Wooldridge, John Wiley, [Cap. 11]  Multiagent Systems. Algorithmic, Game-Theoretic, and Logical Foundations. Y. Shoham, K. Leyton-Brown, Cambridge, [Cap. 3] (Free download at  Trust, Kinship and Locality in the Iterated Prisoner’s Dilemma. M. Venanzi. MSc Thesis. [Cap. 5] (PDF available here)here


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