1. 1 Quantum Games Quantum Strategies in Classical Games Presented by Yaniv Carmeli

2. 2 Talk Outline Introduction  Game Theory  Why quantum games? PQ Games  PQ penny flip 2x2 Games  Quantum strategies

3. 3 Game Theory The study of decision making of competing agents is conflict situations.  Economic problems  Diplomatic relations  Social sciences  Biology  Engineering John von Neumann

4. 4 Why Quantum Games? Attempt to understand the source of the advantages of quantum computation. Quantum algorithms as games.  Which problems are solvable more efficiently using quantum algorithms? Quantum communication as a game.  The objective is to maximize effective communication Quantum cryptography as a game  Eve’s objective is to learn the contents of the conversation.

5. 5 Game Theory - Terminology Players Moves Strategy  Instructions to the player how to react to all scenarios of the game. Pure strategy – Always play a given move. Mixed strategy – Probabilistic choice of moves.

6. 6 Game Theory – Terminology (Cont.) Utility  Numerical measure of the desirability of an outcome. Payoff Matrix  Gives the utility for all the players and for all game outcomes.

7. 7 Game Theory – Terminology (Cont.) A Nash Equilibrium (NE)  A combination of strategies from which no player can improve his payoff by unilateral change of strategy.

8. 8 PQ Coin Flip PQ

9. 9 Coin Flip (Cont.) The Game:  A penny is placed in a box head up.  Q can choose to flip or not to flip.  P can choose to flip or not to flip.  Q can choose to flip or not to flip  At the end: If the coin is head up, Q wins, else P wins.

10. 10 Coin Flip (Cont.) No deterministic solution. Best mixed strategy: Flip with probability ½, don’t flip with probability ½.  Expected payoff: 0. General probabilistic strategy: Flip with probability p, don’t flip with probability 1-p.

11. 11 PQ Coin Flip P Q

12. 12 Coin Flip – Quantum Representation The coin is represented by a qubit, where represents head up, and represents tail.  Initial state : Flipping the coin: Not flipping: Probabilistic strategy:

13. 13 Coin Flip – A Quantum Player A quantum player is allowed any unitary strategy. Q’s first operation is After Picard’s mixed strategy:

14. 14 Coin Flip (Cont.) What if the game was to end here?  If Q were to employ a strategy for which Picard could get an expected payoff of by selecting p=0 (or p=1).  If Picard were to choose Q could get an expected payoff of by selecting a=1 (or b=1)  NE is: Where

15. 15 Coin Flip (Cont.) NE is: where This represent the same results as in a classic game. A quantum player has no advantage if he has only one move, and there is no entanglement involved.

16. 16 Coin Flip (Cont.) Q has a winning strategy:  After Q’s first move:  After P’s move:  After Q’s second move: The mixed/quantum equilibria: with exp. payoff of 1 to Q.

17. 17 Coin Flip (Cont.) What about a game with two quantum players? Consider an arbitrary pair of quantum strategies  If Q can improve his expected payoff by choosing  If P can improve his expected payoff by choosing No Equilibrium!

18. 18 Coin Flip – Bad Example? Can be implemented classically – not an example for superiority of a quantum player (S.J. van Enk, 2000). Classical implementations are not scalable – quantum implementations are (D.A. Meyer, 2000). It’s like losing a game of chess and saying: “If we would have played on a larger board, I would have won”. (S.J. van Enk, 2000).

19. 19 Game Theory – Terminology (Cont.) A Dominant Strategy  Does at least as well as any other strategy against any possible moves by the other players.

20. 20 Game Theory – Terminology (Cont.) A Pareto optimal outcome  An outcome from which no player can increase his utility without reducing the utility of another player.

21. 21 2x2 Games 2 players Each has the choice between 2 pure strategies.  The choice has to be made without communication, and before knowing the opponent’s chosen move.  The payoff matrix is known. Assumptions on the players:  Rationality - Players aim to maximize their payoffs  Each player knows that other players are rational

22. 22 Two suspects held in separate cells are charged with a major crime. However, there is not enough evidence. Both suspects are told the following policy:  If neither confesses - both will be convicted of a minor offense and get one year in jail.  If both confess - both will be sentenced to six years.  If one confesses but the other does not, then the confessor will be released, but the other will be sentenced to jail for nine years. The Prisoners ’ Dilemma

23. 23 Each player has the choice between 2 strategies:  C (Cooperate)  D (Defect) Nash Equilibrium: [D,D] Pareto Optimal Strategy: [C,C] The Prisoners ’ Dilemma

24. 24 PD – Another Version Two firms, Reynolds and Philip, share a market. Each firm earns $60M from its customers if neither do advertising. Advertising costs a firm $20M. Advertising captures $30M from the competitor.

25. 25 PD – Third Version Two cyclists halfway in a race, with the rest of the cyclists far behind them. Each has two options: Taking the lead, where there is no shelter from the wind (C), or staying behind and riding in the other’s slipstream (D). If they both make no effort to stay ahead, the rest of the cyclists will catch up. If one takes the lead, he works much harder and the other cyclist is likely to win.

26. 26 PD – Yet Another Version Two states engaged in an arms race. They both have two options, either to increase military expenditure (D) or to make an agreement to reduce weapons (C). Neither state can be certain that the other one will keep to such an agreement; therefore, they both incline towards military expansion.

27. 27 Each player has the choice between 2 strategies:  C (Cooperate \ Swerve)  D (Defect \ Don’t Swerve) Nash Equilibria: [C,D],[D,C] Pareto Optimal Strategy: [C,C] Chicken

28. 28 Each player has the choice between 2 strategies:  O (Opera)  T (Television) Nash Equilibria: [O,O],[T,T] Pareto Optimal Strategy: [O,O],[T,T] Battle of The Sexes

29. 29 Alice may want a date with Bob, but if he doesn’t want a date with her, she doesn’t want him to know that she was interested. Bob may want a date with Alice, but if she doesn’t want a date with him, he doesn’t want her to know that he was interested. Is there still hope for the shy Alice and Bob? Is this a 2x2 game? The Dating Problem

30. 30 Quantum 2x2 Games The players receive two qubits (one for each) in a known initial state. Each player manipulates his qubit according to his chosen move. At the end, both qubits are measured using a predetermined known basis. The expected payoff is determined according to the payoff matrix:

31. 31 Quantum 2x2 Games (Cont.) Observation: If the qubits’ initial state is not entangled, there is no advantage over a classical player utilizing a mixed strategy. What if the initial state was the maximally entangled state ?

32. 32 Quantum Prisoners ’ Dilemma For the rest of this section we consider the Prisoners’ Dilemma with the initial state: and the basis for measurement: The payoff of final state  :

33. 33 When Alice and Bob select their strategies from S (CL) (local rotations with one parameter):  Note that: One Parameter Set of Strategies

34. 34 The expected payoff: The same as mixed strategies in the classical version. (Like choosing to cooperate with probability ). Equilibrium is still [D,D]. One Parameter Set of Strategies

35. 35 Now Alice and Bob select their strategies from S (TP) (the two-parameter set of operators):  Note that: Two Parameter Set of Strategies

36. 36 The expected payoff: There is a new unique equilibrium [Q,Q], where: Since, it is also the optimal solution (Given without proof). Two Parameter Set of Strategies

37. 37 Proof that [Q,Q] is an equilibrium: If Bob uses Q, then for every strategy of Alice U( ,  ) the same argument holds for every strategy of Bob, when Alice uses Q. Q is unique – given without proof. Two Parameter Set of Strategies

38. 38 If Alice and Bob select their strategies from S (GU) (the set of general local unitary operations): For every move of Bob (where a,b,c,d are appropriate complex numbers), there exists a move for Alice, s.t.. The same argument is true for Bob, so there is no Nash Equilibrium in pure strategies. General Unitary Operations as Strategies

39. 39 There exists a NE in mixed strategies: Expected payoff for both is 2.5. General Unitary Operations as Strategies

40. 40 Problem: It is not the only equilibrium.  For example: The following is also an equilibrium It has the same property as the previous one.  And there are more… Which of them will be chosen? General Unitary Operations as Strategies

41. 41 There exist probabilities and unitary operators s.t.: If Alice will choose “R”, Bob’s actions will not change the state of the quantum system anymore. [R,R] is the single NE which is the only one that gives an expected payoff of 2.25 for both players General Unitary Operations as Strategies

42. 42 Focal points are outcomes which are distinguished from others on the basis of some characteristics which are not included in the formalism of the model. Those characteristics may distinguish an outcome as a result of some psychological or social process and may even seem trivial, such as the names of the actions. If there are more that one NEs and one of them can be considered a focal equilibrium, then it is the one that will be chosen. The NE is [R,R] General Unitary Operations as Strategies

43. 43 Can we say we have cracked the prisoners’ dilemma? The solution [D,D] is not a NE anymore! The effect of Q can be described classically:  If Alice chooses Q and Bob chooses C or D then his choice is changed to the one he didn’t pick (and vice versa).  If both choose Q – the payout is as if both cooperated. Is It Really That Good?

44. 44 In non-cooperative games players are not allowed to communicate, and the use of correlated random variables is not allowed. Letting Alice and Bob use an entangled state, means they are using the correlations present in such a state. This goes against the spirit of the prisoners’ dilemma. Is It Really That Good?

45. 45 2x2 Games With Entanglement Before each player makes his move we apply an entangling operation to the qubits. We would like the classic game to be a sub-game of the quantum one. Hence:  After the players make their moves, we will apply.  commutes with any direct product of any pair of classical strategies.

46. 46 2x2 Games With Entanglement With the entangling operator: we can control the level of entanglement.  - No entanglement.  - Maximally entangled state. Is there a critical level of entanglement, below which the quantum player has no advantage over the classical player?