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Multiagent Interaction

Multiagent Interaction. Mohsen Afsharchi. What are Multiagent Systems?. We are…. Self-interested agents It does not mean that: they want to harm other agents they only care about things that benefit them It means that

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Multiagent Interaction

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  1. Multiagent Interaction MohsenAfsharchi

  2. What are Multiagent Systems?

  3. We are… • Self-interested agents • It does not mean that: • they want to harm other agents • they only care about things that benefit them • It means that • the agent has its own description of states of the world that it likes, and that its actions are motivated by this description

  4. Utilities and Preferences • Assume we have just two agents: Ag = {i, j} • Agents are assumed to be self-interested: they have preferences over how the environment is • Assume W = {w1, w2, …}is the set of “outcomes” that agents have preferences over • We capture preferences by utility functions:ui =W Ruj =W R • Utility functions lead to preference orderingsover outcomes:w w’ means ui (w) > ui (w’)

  5. What is Utility? • Utility is not money (but it is a useful analogy) • Typical relationship between utility & money:

  6. Multiagent Encounters • We need a model of the environment in which these agents will act… • agents simultaneously choose an action to perform, and as a result of the actions they select, an outcome in W will result • the actualoutcome depends on the combinationof actions • assume each agent has just two possible actions that it can perform, C (“cooperate”) and D (“defect”) • Environment behavior given by state transformer function:

  7. Multiagent Encounters • Here is a state transformer function:(This environment is sensitive to actions of both agents.) • Here is another:(Neither agent has any influence in this environment.) • And here is another:(This environment is controlled by j.)

  8. Rational Action • Suppose we have the case where bothagents can influence the outcome, and they have utility functions as follows: • With a bit of abuse of notation: • Then agent i’spreferences are: • “C” is the rational choicefor i. (Because iprefers all outcomes that arise through C over all outcomes that arise through D.)

  9. Payoff Matrices • We can characterize the previous scenario in a payoff matrix: • Do we need to engage in strategic thinking? • Pay-off matrix in fact defines a game (technically strategic game)

  10. Game Theory… • The study of games! • Bluffing in poker • What move to make in chess • How to play Rock-Scissors-Paper • Also • Also study of auction design, strategic deterrence, election laws, coaching decisions, routing protocols,...

  11. Game theory… • Game theory is a formal way to analyze interactionsamong a group of rationalagents who behave strategically. • Group: Must have more than one decision maker • Otherwise you have a decision problem, not a game • Interaction: What one agent does directly affects at least one other agent • Strategic: Agents take into account that their actions influence the game • Rational: An agent chooses its best action (maximizes its expected utility)

  12. Game in Normal Form

  13. Game in Normal Form…

  14. Strategy

  15. Strategy…

  16. Solution Concepts

  17. Dominant Strategies • Given any particular strategy (either C or D) of agent i, there will be a number of possible outcomes • We say s1dominatess2if every outcome possible by iplaying s1is preferred over every outcome possible by iplaying s2 • A rational agent will never play a dominated strategy • So in deciding what to do, we can delete dominated strategies • Unfortunately, there isn’t always a unique undominated strategy

  18. Dominant Strategies… • Iterated elimination of strictly dominated actions (IESDA) is a solution technique that iteratively eliminates strictly dominated actions from all agents, until no more actions are strictly dominated.

  19. Nash Equilibrium • In general, we will say that two strategies s1 and s2are in Nash equilibrium if: • under the assumption that agent iplays s1, agent jcan do no better than play s2; and • under the assumption that agent jplays s2, agent ican do no better than play s1. • Neither agent has any incentive to deviate from a Nash equilibrium • Unfortunately: • Not every interaction scenario has a Nash equilibrium • Some interaction scenarios have more than one Nash equilibrium

  20. Nash Equilibrium There is no dominant strategy. Moreover there is no Nash Equilibrium. Whichever cell you choose, at least one of you would have preferred to make another choice, assuming the other player did not changed their choice.

  21. Mixed Strategy • Introducing Randomness… • If player has k possible choices, s1 , s2 ,…, skthen the mixed strategy over these choices takes the form • Play s1 with probability p1 • Play s2 with probability p2 • … • Play sk with probability pk • Where p1 + p2 +…+ pk=1 • Every game in which every player has a finite set of possible strategies has a Nash equilibrium in mixed strategies

  22. Matching Pennies game • Each of two agents chooses either Head or Tail. If the choices differ, agent 1 pays agent 2 a cent; if they are the same, agent 2 pays agent 1 a cent. • There is no (pure strategy) Nash equilibrium in this game. If we play this game, we should be “unpredictable.” • That is, we should randomize (or mix) between strategies so that we do not get exploited.

  23. Matching Pennies game… • But not any randomness will do: Suppose Player 1 plays .75 Heads and .25 Tails (that is, Heads with 75% chance and Tails with 25% chance). Then Player 2 by choosing Tails (with 100% chance) can get an expected payoff of 0.75×1 + 0.25×(-1) =0.5. But that cannot happen at equilibrium since Player 1 then wants to play Tails (with 100% chance) deviating from the original mixed strategy. • Since this game is completely symmetric it is easy to see that at mixed strategyNash equilibrium both players will choose Heads with 50% chance and Tails with 50% chance. • In this case the expected payoff to both players is 0.5×1 + 0.5×(-1) = 0 and neither can do better by deviating to another strategy.

  24. Pareto Efficiency • Is not really a solution concept. • An outcome is Pareto efficient if there is no other outcome that improves one player’s utility without making somebody else worse off. • Battle of the sexes

  25. Maximizing Social Welfare • We measure how much utility is created by an outcome in total. SW(w) Becomes relevant if all the agents within a system have the same owner

  26. Competitive and Zero-Sum Interactions • Where preferences of agents are diametrically opposed we have strictly competitivescenarios • Zero-sum encounters are those where utilities sum to zero:ui (w) +uj (w) = 0 for all wW • Zero sum implies strictly competitive • Zero sum encounters in real life are very rare … but people tend to act in many scenarios as if they were zero sum

  27. The Prisoner’s Dilemma • 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 three 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

  28. The Prisoner’s Dilemma • Payoff matrix forprisoner’s dilemma: • Top left: If both defect, then both get punishment for mutual defection • Top right: If icooperates and jdefects, igets sucker’s payoff of 1, while jgets 4 • Bottom left: If jcooperates and idefects, jgets sucker’s payoff of 1, while igets 4 • Bottom right: Reward for mutual cooperation

  29. The Prisoner’s Dilemma • The individual rationalaction is defectThis guarantees a payoff of no worse than 2, whereas cooperating guarantees a payoff of at most 1 • So defection is the best response to all possible strategies: both agents defect, and get payoff = 2 • But intuitionsays this is notthe best outcome:Surely they should both cooperate and each get payoff of 3! • We have better choice (?) and it is not rational. Why?

  30. The Prisoner’s Dilemma • This apparent paradox is the fundamental problem of multi-agent interactions.It appears to imply that cooperationwill not occur in societies of self-interested agents. • Real world examples: • nuclear arms reduction (“why don’t I keep mine. . . ”) • free rider systems — public transport; • The prisoner’s dilemma is ubiquitous. • Can we recover cooperation?

  31. Arguments for Recovering Cooperation • Conclusions that some have drawn from this analysis: • the game theory notion of rational action is wrong! • somehow the dilemma is being formulated wrongly • Arguments to recover cooperation: • We are not all Machiavelli! and not all games are PD • Yes but there are always punishments !!! • The other prisoner is my twin! • Then I am damn fool to behave any other way (Because it is not a game) • The shadow of the future…

  32. The Iterated Prisoner’s Dilemma • One answer: play the game more than once • If you know you will be meeting your opponent again, then the incentive to defect appears to evaporate • Cooperation is the rational choice in the infinititely repeated prisoner’s dilemma(Hurrah!)

  33. Backwards Induction • But…suppose you both know that you will play the game exactly ntimes • On round n - 1, you have an incentive to defect, to gain that extra bit of payoff… • But this makes round n – 2the last “real”, and so you have an incentive to defect there, too. • This is the backwards inductionproblem. • Playing the prisoner’s dilemma with a fixed, finite, pre-determined, commonly known number of rounds, defection is the best strategy

  34. Axelrod’s Tournament • Suppose you play iterated prisoner’s dilemma against a rangeof opponents…What strategy should you choose, so as to maximize your overall payoff? • Axelrod (1984) investigated this problem, with a computer tournament for programs playing the prisoner’s dilemma

  35. Strategies in Axelrod’s Tournament • ALLD: • “Always defect” — the hawkstrategy; • TIT-FOR-TAT: • On round u = 0, cooperate • On round u > 0, do what your opponent did on round u – 1 • TESTER: • On 1st round, defect. If the opponent retaliated, then play TIT-FOR-TAT. Otherwise intersperse cooperation and defection. • JOSS: • As TIT-FOR-TAT, except periodically defect

  36. Recipes for Success in Axelrod’s Tournament • Axelrod suggests the following rules for succeeding in his tournament: • Don’t be envious:Don’t play as if it were zero sum! • Be nice:Start by cooperating, and reciprocate cooperation • Retaliate appropriately:Always punish defection immediately, but use “measured” force — don’t overdo it • Don’t hold grudges:Always reciprocate cooperation immediately

  37. Game of Chicken • The game of chicken:(Swerving = coop, Driving straight = defect.) • Difference to prisoner’s dilemma:Mutual defection is most feared outcome.(Whereas sucker’s payoff is most feared in prisoner’s dilemma.) • Strategies (C,S) and (S,C) are in Nash equilibrium

  38. Other Symmetric 2 x 2 Games • Given the 4 possible outcomes of (symmetric) cooperate/defect games, there are 24 possible orderings on outcomes • CC ši CD ši DC ši DDCooperation dominates • DC ši DD ši CC ši CDDeadlock. You will always do best by defecting • DC ši CC ši DD ši CDPrisoner’s dilemma • DC ši CC ši CD ši DDChicken • CC ši DC ši DD ši CDStag hunt

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