Quantum games


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My Presentation on Quantum Games during a course at IIT Kanpur. This is just to get a feeling for my 1st presentation on slideshare :)

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Quantum games

  1. 1. Quantum Games and Quantum  Strategies The Future of Decision Making ?
  2. 2. Quantum Games !!!● Can principles of Quantum communication be used to develop efficient and unbiased marketing strategies ?● Quantum communication and Quantum cryptography can be regarded as Games played between 2 legal players Alice and Bob and the illegal players in between them trying to decrypt the secret message.
  3. 3. What is a Game ?● We have a set of players i● a strategy set Si for each player i.● Pay-off for each player Pi(s1,s2,....), where siSi.
  4. 4. Where does “Quantum” comes into “play”?● Game theory does not explicitly concern itself with how the information is transmitted once a decision is taken. Bearing in mind that a game is also about the transfer of information, it becomes legitimate to ask what happens if these carriers of information are taken to be quantum systems, quantum information being a fundamental notion of information.
  5. 5. Strategies and Equilibriums● A quantum strategy sA is called dominant strategy of Alice if PA (sA,sB) ≥ PA (s′A, s′B ) for all s′A ∈ SA , s′B ∈ SB● A pair ( sA , sB ) is said to be an equilibrium in dominant strategies if sA and sB are the players’ respective dominant strategies
  6. 6. Strategies and Equilibriums● A combination of strategies ( sA , sB ) is called a Nash equilibrium if PA (sA,sB) ≥ PA (s′A, sB ) and PB (sA,sB) ≥ PB (sA, s′B )for all s′A ∈ SA , s′B ∈ SB .● A pair of strategies ( sA , sB ) is called Pareto optimal, if it is not possible to increase one player’s pay-off without lessening the pay-off of the other player.
  7. 7. Nash Equilibrium● A Nash equilibrium implies that neither player has a motivation to unilaterally alter his or her strategy from this equilibrium solution, as this action will lessen his or her pay-off. Given that the other player will stick to the strategy corresponding to the equilibrium, the best result is achieved by also playing the equilibrium solution.
  8. 8. The Prisoners Dilemma● Two prisoners are being questioned by the police. They are held in separate cells and cannot talk to each other. The police make the following offer to both the prisoners: if one confesses that both committed the crime then the confessor will be set free and the other will spend 5 years in jail; if both confess then they will each get 4 yr jail term: if neither confess, then they will each spend 2 yrs in jail
  9. 9. The Prisoners Dilemma● Table of Pay-offs : P1 Cooperation Defection P2 Cooperation (3,3) (0,5) Defection (5,0) (1,1)
  10. 10. The Prisoners Dilemma● If both the prisoners co-operate with each other then they are each awarded 3 yrs of freedom, if none of them co-operate with each other then each will be awarded 1 yr of freedom and if one of them has conscience and the other is defected then the good prisoner has to serve full 5 yr in jail while the bad one will be set free immediately i.e. Full 5 yrs of freedom for the bad prisoner. So what is the optimal strategy that both of them apply here?
  11. 11. Nash Equilibrium● In the Prisoners dilemma problem if the dominant strategy is (D,D) with pay-offs (1,1) for Alice and Bob. Now if Alice changes her strategy while Bob sticks to his strategy, i.e. Alice decides to become good then she pays the price with 0 pay-off or in this case a complete 5yr jail sentence. Same is true for Bob, thus if either of them decides to become good while the other stays bad, the his/her pay-off decreases, hence (D,D) is in Nash equilibrium. But (D,D) is not Pareto optimal because (C,C) has better pay-offs than (D,D).
  12. 12. Quantum Prisoners Dilemma● In traditional 2 × 2 games where each player has just a single move, creating a superposition by utilizing a quantum strategy will give the same results as a mixed classical strategy. In order to see non- classical results it is necessary to produce entanglement between the players’ moves● Initial state is the maximally entangled state |> = (|00> + i|11>)/√2
  13. 13. Quantum Prisoners Dilemma● The final state is represented as following : |ψf> = J†(UA ⊗UB) J |CC> where J=exp(iγD ⊗ D/2), γ ∈ [0, π/2] is a real parameter, UA and UB are respective strategies for Alice and Bob.● Expected Pay-off for Alice : <$> = ACC|<ψf |CC>|2 + ACD|<ψf |CD>|2 + ADC| <ψf |DC>|2 + ADD|<ψf |DD>|2
  14. 14. Quantum Prisoners Dilemma● The matrix representation of operators corresponding to quantum strategies from this set is given by U(,) = {( eicos(/2), sin(/2) ), ( -sin(/2), e-icos(/2) )}● The strategies are : C={(1,0),(0,1)} and D={(0,1),(-1,0)}● γ is a measure for the game’s entanglement
  15. 15. Pay-Offs is what matters !!● Alices expected pay-off with the quantum strategies: PA (θA , φA , θB , φB ) = 3 |cos(φA + φB ) cos(θA /2) cos(θB /2)|2 + 5 |sin(φA ) cos(θA /2) sin(θB /2) − cos(φB ) cos(θB /2) sin(θA /2)| 2 + |sin(φA + φB ) cos(θA /2) cos(θB /2) + sin(θA /2) sin(θB /2)|2 .
  16. 16. Pay-Offs is what matters !!● Bobs expected pay-off with the quantum strategies: PB (θA , φA , θB , φB ) = 3 |cos(φA + φB ) cos(θA /2) cos(θB /2)|2 + 5 |sin(φB ) cos(θB /2) sin(θA /2) − cos(φA ) cos(θA /2) sin(θB /2)| 2 + |sin(φA + φB ) cos(θA /2) cos(θB /2) + sin(θA /2) sin(θB /2)|2 .
  17. 17. Pay-Offs is what matters !!● Assuming Bob chooses D= U (π, 0), then PA (θA , φA ,  , 0 ) = 5 |sin(φA ) cos(θA /2)| 2 + |sin(θA /2)|2 <= 5 for θA = 0 and φA = /2● Thus Alice’s best reply would be Q = U(0, /2) = {(i,0),(0,-i)}● While assuming Bob plays C= U (0, 0) Alice’s best strategy would be defection D. Thus, there is no dominant strategy left for Alice. The game being symmetric, the same holds for Bob, i.e., D ⊗ D is no longer an equilibrium in dominant strategies
  18. 18. Quantum Nash Equilibrium● PA (θA , φA , 0 , /2 ) = 3 |sin(φA) cos(θA /2) |2 + |cos(φA) cos(θA /2)|2 <= 3 for θA = 0, φA = /2 thus PA (U(θA , φA ) , Q ) <= PA (Q, Q )● PB (0 , /2, θB , φB ) = 3 |sin(φB)cos(θB /2)|2 + |cos(φB) cos(θB /2)|2 <= 3 for θB = 0, φB = /2 thus PB (Q, U(θB , φB ) ) <= PB (Q, Q )● Hence (Q,Q) is the new Nash Equilibrium.
  19. 19. Does the Prisoners escape the dilemma?● It is interesting to see that Q ⊗ Q has the property to be Pareto optimal , that is, by deviating from this pair of strategies it is not possible to increase the pay-off of one player without lessening the pay-off of the other player. In the classical game only mutual cooperation is Pareto optimal, but it is not an equilibrium solution. One could say that by allowing for quantum strategies the players escape the dilemma .
  20. 20. Real Life Dilemmas !!● Nuclear proliferation among nations: When a nation tries to up its nuclear armory and sources with the help of its more powerful allies, its rival nations feels threatened and they too power up their nuclear warfares and sources with powerful allies, thus increasing tension between nations and increasing possibility of nuclear war. Thus one might speculate that Governments of nations using Quantum strategies might even end the possibility of a nuclear disaster in near future, but then “everything” needs to be “Quantum” !!!
  21. 21. Real Life Dilemmas !!This ones from “Batman-Dark Knight” : Jokerplanted bombs on the 2 ferries where the triggers forthe bombs were given to the opposite parties. Theclassical Nash equilibrium strategy would lead the 2parties to trigger the bombs simulteneously thuskilling everybody. But in the movie they might havefigured out the “Quantum strategy” to co-operatethus saving all of them. One might think “Quantumstrategies” as the most “socially acceptable” or“near conscience” but wait till you have seen theopposite face of the “coin”.
  22. 22. Real Life Dilemmas !!● Breakdown of talks at the Climate change Summit at Copenhagen: Rich nations having larger number of industries are emmitting larger amount of CO2 in the atmosphere. But only developing nations trying to build up its commercial and industrial economy are asked to cut down CO2 emmision. Such nations felt that the rich are trying to suppress them down and they deny to decrease CO2 emmision from factories etc. and thus talks broke down. Defection( increased release of CO2 by all nations) seemed the Nash equilibrium strategy compared to Cooperation(decreased release of CO2 by all nations). Thus a “Quantum World” should be the paradise or the heaven to live in.
  23. 23. Lets Make A Deal !!
  24. 24. Should you exchange doors ?● The answer is yes if you are a rational person !● You win 2/3 of the time you exchange the doors and lose 1/3 of the time !!● From an information theoretic point of view, by opening a door without a prize Monty has given information about where the prize is. Lets see how?
  25. 25. Should you exchange doors ?● The probability of originally choosing a goat is 2/3 and the probability of originally choosing the car is 1/3. Once Monty Hall has removed a "goat door," the contestant who chose the door with a goat behind it will necessarily win the car, and the contestant who originally chose the car will necessarily "win" the goat. Because the chances are 2/3 of being a contestant who originally chose a goat, probability will always favor switching choices.
  26. 26. Should you exchange doors ?
  27. 27. Quantum Monty Hall !!● In our quantum version of the game let us call the host as Alice and player as Bob.● There is one quantum particle and three boxes |0 > , |1> , and |2>● The state of the system can be expressed as |> = |oba> where a = Alice’s choice of box( prize door ), b =Bob’s choice of box, and o = the box that has been opened
  28. 28. Quantum Monty Hall !!● The final state of the system is |f> = (S cos  + N sin )O(I  B  A)|i>● Where A Alice’s choice operator or strategy, B Bob’s initial choice operator or initial strategy, O the opening box operator, S Bob’s switching operator, N Bob’s not switching operator, I the identity operator, and  = [0,/2]
  29. 29. Quantum Monty Hall !! The open box operator is a unitary operator that can be written as : O = ∑ijkl |∈ijk| |njk><ljk| + ∑jk |mjj><ljj|● where |∈ijk| = 1 if i,j,k are different else 0. m = (j+l+1) (mod 3) and n = (i+l)(mod 3).● The first term considers the cases where Bob chooses the door not having the car behind it and the second term considers the cases where Bob initially chooses the door having the car inside it
  30. 30. Quantum Monty Hall !!● The possible cases are : |010> ⇒|210>, |020> ⇒|120>, |001> ⇒|201>, | 021> ⇒|021>, |002> ⇒|102>, |012> ⇒|012>● Thus the operator should be |210>< 010| + |120>< 020| + |201>< 001| + |021>< 021| + |102>< 002| + |012><012| to introduce unitarity we use the complete operator ∑ijkl |∈ijk| |njk><ljk|
  31. 31. Quantum Monty Hall !!● Bobs switch operator can be written as : S = ∑ijkl |∈ijk| |ikk><ijk| + ∑ji |imj><ijj| m=(3-i-j)● The possible transformations we consider as probable : |120> ⇒|100>, |100> ⇒|120>, |011> ⇒|021>, |201> ⇒|211>, |021> ⇒|011>, |122> ⇒|102>, | 022> ⇒|012>, |012> ⇒|022>, |102> ⇒|122>, |200> ⇒|210>, |210> ⇒|200>, |211> ⇒|201>.
  32. 32. “Expectation”● Bob wins if his choice of door is same as the door Alice chooses to keep the car. Thus expectation value of Bobs win = <$B> = ∑ij |<ijj|f>|2● Alice wins if Bob is incorrect, so <$A> = 1 - ∑ij |<ijj|f>|2
  33. 33. Quantum Monty Hall !!● Without entanglement and one of Alice or Bob applying the identity operator we get back the classical case where Bob wins with 2/3 probability if he switches doors.● Consider the case with maximum entanglement The initial state is |i> = |0>  (1/3)(|00> + |11> + |22>)
  34. 34. Quantum Monty Hall !!● O(IBA)|i> = (1/3) ∑ijkl |∈ijk| bljalk|ijk> + (1/3) ∑jl bljalj|mjj>● SO(IBA)|i> = (1/3) ∑ijkl |∈ijk| bljalk|ikk> + (1/3) ∑jkl |∈jkm| bljalj|mkj>● If B=I then <$B> = (1/3)sin2 (|a00|2 + |a11|2 + |a22|2) + (1/3)cos2 (|a01|2 + |a02|2 + |a10|2 + |a12|2 + |a20|2 + |a21|2)
  35. 35. Quantum Monty Hall !!● If now Alice chooses an unitary operator whose diagonal element all have absolute value (1/ 2) and off-diagonal elements have absolute value (1⁄2) then : <$B> = (1⁄2) sin2 + (1⁄2) cos2 hence payoff for Bob if he switches = 1⁄2 .
  36. 36. Is the Game Fair Now ?● For the Monty Hall game where both participants have access to quantum strategies, maximal entanglement of the initial states produces the same payoffs as the classical game. That is, for the Nash equilibrium strategy the player, Bob, wins two-thirds of the time by switching boxes. If the host, Alice, has access to a quantum strategy while Bob does not, the game is fair, since Alice can adopt a strategy with an expected payoff of 1/2 for each person, while if Bob has access to a quantum strategy and Alice does not he can win all the time.
  37. 37. Simple Penny Flipover● There are 2 players involved in the penny flip game. Lets name them P and Q. P is to place a penny, head up, in a box, whereupon they will take turns (Q, then P, then Q) flipping the penny over (or not), without being able to see it. Q wins if the penny is head up when they open the box● Q ---- NN NF FN FF P N -1 1 1 -1 F 1 -1 -1 1
  38. 38. Simple Penny Flipover● Suppose P doesn’t flip the penny over. Then if Q flips it over an even number of times, P loses. Similarly, if P flips the penny over, then if Q flips it over only once, P loses. Thus PQ penny flipover has no deterministic solution , no deterministic Nash equilibrium : there is no pair of pure strategies, one for each player, such that neither player improves his result by changing his strategy while the other player does not.
  39. 39. Simple Penny Flipover● Since this is a two-person zero-sum strategic game with only a finite number of strategies, there is a probabilistic solution: It is easy to check that the pair of mixed strategies consisting of P flipping the penny over with probability 1⁄2 and Q playing each of his four strategies with probability 1⁄4 is a probabilistic Nash equilibrium: Neither player can improve his expected payoff (which is 0 in this case) by changing the probabilities with which he plays each of his pure strategies while the other player does not.
  40. 40. Simple Penny Flipover● In the classical case both P and Q has a 1⁄2 probability of winning the game but if Q adopts a quantum strategy then she will win with a probability of 1● |0> = |H> then Q: |1> = H|0> = (1/2)(|H> +|T>) then P(mixed classical strategy) : U= pF + (1-p)N, |2> = U|1> = (1/2)(|H> + |T>), then again Q: |3> = H|2> = |H>
  41. 41. Penny Flipover Applications● This strategy is same as that of the game between the oracle and the Grovers algorithm:● In the problem of searching a database of size N, the locations in the database correspond to pure strategies. The oracle can be thought of as the player P in the game who uses a mixed strategy to frustate our search for an item at some specified location. While the Grovers algorithm represents the player Q who uses quantum strategy.
  42. 42. Penny Flipover Applications● Even the Shors algorithm can be interpreted as a penny flipover game between a Quantum player and a classical mixed strategist. After we apply the inverse Quantum Fourier Transform, we measure a certain value as the output measurement, then we use classical continued fraction to find out the period accurately with some probability. The continued fraction implementation can be thought of as the player P who uses mixed strategy to produce a correct result with some probability.
  43. 43. Penny Flipover● One might ask what if player P also uses a Quantum strategy?● In the penny flipover game P using the Quantum operator U = √p X + ie- (1-p) Z where X is the Pauli NOT gate and  is the phase, would produce the same result as before with Q winning with probability of 1.
  44. 44. Penny Flipover● Theorem : A two person zero sum game need not have a (quantum, quantum) equilibrium.● Proof : Let us assume that the quantum strategies [(U2),(U1,U3)] is a (quantum, quantum) equilibrium. Suppose that U3 U2 U1 |H>  |H>. Then Q can improve his expected payoff to 1 by changing his strategy U3 to U1-1U2-1 which is unitary. Also suppose U3 U2 U1 |H>  |T> then P can improve his expected payoff to 1 by changing his strategy U 2 to U3-1XU1-1 which is also unitary.
  45. 45. Concluding Remarks● One might speculate that game theoretic perspective will suggest new possibilities for efficient quantum algorithms.
  46. 46. Concluding Remarks● It may be too soon to say that Quantum game theory and quantum strategies are going to be the future of decision making in fields like Finance, Marketing, Adminstrative Policies, Quantum Algorithms, Quantum communication(cryptography) and everyday life decision processess, given that efficient hardware implementation of quantum computation is still an uphill task with the noise and decoherence effects playing their parts
  47. 47. Still a Hope !!● The day when the Quantum error correction codes becomes so efficient that all noises and decoherences can be cancelled out, “Quantum” will be the order of the day, and the world will take the next leap forward changing gears from the “Classical” to “Quantum”. I strongly believe that the day is not far away and it will change the whole world, “Logic and reasoning will dominate over emotions and feelings”.
  48. 48. Thank you !