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The corresponding bimatrix for the prisoners game is 1. The number on the left is for the years the prisoner S1 prisoner must serve. Quantum game theory is a classic game theory generalization. That is, quantum game strategies and outcomes include the classical as particularities, but also quantum features let the application of new strategies which leads to solutions classically imposible. Quantum games provide new ways to cooperate, to eliminate dilemmas, and as a consequence new equilibriums arise.
That is, the system equilibrium is not longer C,C to be N,N.
Unlike the traditional game, in the quantum version of the dating game, players get the chance to use quantum techniques, for example they can explore their possibilities using a quantum search algorithm. That state space must be capable of being translatable, say to a graph G where to find some particular state which has a searched feature or distinctive mark, throughout the execution of the algorithm.
Furthermore it is possible to guarantee that the searched node is marked by a minimum maximum value of a physical property included in the algorithm. Let agents be coded as Hilbert space base states. Table 1 displays four women states in the first column and some feature that makes them unique in the second column which we will code with a letter for simplicity. Left column contains women states and right column displays a letter representing some feature or a feature set that characterizes each woman on the left.
As a consequence any quantum algorithm can be thought as a set of suitable linear transformations. Initially, the woman identified by state 0i is chosen with probability one. The next step is to create superposition states and like many other quantum algorithms Grover uses Hadamard transform to do this task since it maps n qubits initialized with 0i to a superposition of all n orthogonal states in the 0i, 1i. The value of f w on a superposition of every possible input w may be obtained .
As a result, the corresponding mathematical expression is: As we already stated our algorithm is based on the classical Gale-Shapley GS algorithm which assigns the role of proposers to the elements of one set, the men say, and of judges to the elements of the other. Actually, for a more symmetric formulation of the algorithm where both sets are, at the same time, proposers and judges, it would be necessary another oracle which evaluates women features matching by means of another function g x , but we will not go into that.
Oracle operator U f makes one of two central operations comprising of a whole operation named Grover iterate G Fig. UR and U ftogether with Hadamard transformations represented by H blocks 1in the order depicted by Fig.
The symbol I in UR equation is the identity operator. In Figure 1 Grover iterate is shown and Grover quantum algorithm scheme is depicted in 2.
As the number of iterations the algorithm makes depends on the size of the options set, this must be known at the beginning of simulations. Every operator has its matrix representation to be used in simulations. We suppose the player chooses a woman who has some specific particularity that would distinguish her from any other of the group, so we construct matrix U f and other matrixes for that purpose.
The evolution of the squared amplitude with the iteration number is shown in Figure 3. The fast increasing of the probability to find the preferred state on each iteration contrasts with the decreasing of the probability to find every other state.
Thus when a given man who wants to date a Nw size set selected woman, he must set his own U f operator out, according to his preferences, and then let the algorithm do the job.
The case of Nm men may be obtained generalizing the single man case: Nevertheless, achieving top choice is hard because of competition from other players and your dream partner may not share your feelings. If all players play quantum, the time to find woman is not an issue and the N stable solutions will be the same as for the classic formulation .
Quantum vs classic To compare the quantum approach efficiency with the classical one we will consider some players playing quantum and others playing classic. Let us follow the evolution of agents representative from each group, Q and C respectively. On the other hand, the only way C has to search such a database is to test the elements sequentially against the condition until the target is found. Two different games where both men want to date with the same woman are presented: In the first one player Q gives player C the chance to play first and both have only one attempt per turn, which means only one question to the oracle.
For the last case we analyzed two alternatives for the classic player: The player who invites the chosen woman first has more chances to succeed, as well as that who asks the same woman more times.
Nevertheless the woman has the last word, and therefore the dating success for each player depends on that woman preferences. So, let us define Pci as the probability that woman i accepts dating the classic player C and Pqi as the probability that she accepts the quantum player Q proposal.
In the next step the Oracle marks one of the prospective women state according men preferences. For a two women set Q uses only one step, but C needs two steps to find the right partner. Winning conditions improve for the quantum player for increasing Nwbut not in a monotonous way, because the number of steps used by Grover algorithm in Q search is an integer that increases in discrete steps.
Under the first game conditions both players have only one attempt by turn. On the other hand player Q, using Grover algorithm as his strategy, can modify states amplitudes in order to increase Quantum Dating Market Quantum Dating Market 9 http: One attempt for both players.
Q outperforms C in all shown cases. The small region where C prevails is not shown. Figure 4 shows that situation outcomes for different Pci and Pqi combinations. After each C attempt the system is forced to collapse to one base state, so a third party, that could be the oracle, arrange the states again and mark the solution.
This implies that player C outperforms player Q. Classic player C has four tries while Q has only one. Section discussion In this section we have introduced a quantum formulation for decision matching problems, specifically for the dating game. This is a highly time consuming task that takes a O N runtime for a classical probabilistic algorithm, being N the women database size.
As a consequence, if every man uses quantum strategy, no one does better than the others, and stability is quickly obtained. The performances of quantum vs. Winning conditions improve for the quantum player for increasing N and the same number of attempts, but not in a monotonous way.
The comparison between quantum and classic performances shows that for the same numbers of attempts, the quantum approach outperforms the classical approach. If the game is set in order that the classic player has N2 opportunities and the quantum player only one, the former player begins to have an advantage over the quantum one when his probability to be accepted by the chosen woman is much higher than the probability for the quantum player.
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Stability of couples There is a group of Nm men and Nw women playing the game. As a result each man has been assigned log2 Nw qubits in order to identify each woman. Note that the S M extends to any possible combination of men elections. On the other side there are the women who receive men proposals and must decide whether to accept or not one of them.
Accordingly, state space of the couples emerge from the Kronecker product of the men and N women spaces,i. That is, U M is applied by men to the qubits that identify the women states, meanwhile the women action on the qubits that identify men states is given by UW. Despite the complete system can be represented by its state vector, when it comes to mixed states the density matrix is more suitable.
It was introduced by von Neumann to describe a mixed ensemble in which each member has assigned a probability of being in a determined state.
From the density operator we can construct and understand the statistical behavior about our system by using statistical mechanics and a criterion of maximum or minimum entropy. As women have the last decision, they must evaluate men proposals and decide to accept one of them or reject all. Equation 9 shows the women density matrix which has no off diagonal elements.
Index i is a four qubits number, the first two qubits represent men 0 and 1 choices respectively and the Quantum Dating Market Quantum Dating Market 13 http: Quantum entropy corresponding to the situation where player 0 varies the probability p to apply strategy U00 other two are the two women possible selections, then 16 are the possible couples states. For example, the state i corresponds to the case that man 0 chooses woman 0 and she accepts him and the same occurs with man 1 and woman 1.
As the game progress, probability amplitudes associated with mismatches must decrease, that because it is considered that people prefer to be coupled. The general formula of Ui is 10, that are rotation operators, as explained in  any qubit operation can be decomposed as a product of rotations. The strategies operators used in the examples are defined below, equations 11 and 12 are applied by man 0. Quantum entropy corresponding to the situation where player 1 varies the probability p to apply strategy U01 state 0i into states that are linear superpositions of 0 and 1, representing states with different probabilities of choosing one woman or the other.
On the other hand, strategies applied by man 1 are presented in 13 and Figures 6 and 7 show two situations where the system entropy varies considerably as a function of the strategies the players use. This situation is completely unstable because it is impossible for the Quantum Dating Market Quantum Dating Market 15 http: This correspond to minimum entropy as can be easily seen in Fig.
Depending on the strategies applied by men, the whole system entropy, that is the couple system entropy changes reaching maximum and minimum limits. As p increases, the mixing of the strategies also increases producing an increase in entropy that indicates a tendency to stability. The mixing of the strategies means that the men proposals are less focused on one woman. A result not shown in the figures is that entropy maxima increase when women preferences are the same for every men.
In this way, maxima and minima entropy points may be used to find stable states. Nevertheless, these stable states may not correspond to equilibria states of the game, because the players utilities has not been considered. In order to find Nash equilibria states, these utilities must be considered.
This is beyond this chapter goals. Maxima and minima entropy points are used to find characteristic strategies that lead to stable and unstable states. Nevertheless, in order to find Nash equilibria states, the players utilities must be considered.
Maxima and minima entropy do not depend only on the strategies of men but also on women preferences, reaching the highest value when they have no preferences, that is when they choose every man with equal probability. On the other hand, minimum entropy correspond to men betting all chips to a single woman, without giving a chance to other woman.
The quantum formulation, which was presented in previous section, proceeds by assigning one basis of a Hilbert state space to each woman. On the other side of the market, women receive men proposals and must decide which is the best, accept it and reject the others. The same reasoning corresponds to women states.
On the other hand, if there is some relationship between two or more men, their actions are non-locally correlated, that is, their decisions are far from being independent. Therefore, we will study the case with correlation between agents by means of quantum entanglement, in other words, how harmful or beneficial can be for players knowing each other in advance.
As we mention in the previous section, players strategies are represented by unitary operators in quantum games. In order to understand the problem we analyze here a simple example of two men and two women. Since any qubit operation can be decomposed as a product of rotations, strategies combinations and possible outcomes are infinite.
As a consequence, focusing on men relationship, we study three relevant cases. As men states are entangled, it is not possible to uncouple their Quantum Dating Market Quantum Dating Market 17 http: Since there is no way that men choose the same woman it is a state of mutual cooperation.
Following , equation 16 represents man 0 payoff, where P00 and P01 are his chances to be accepted for a date with woman 0 and 1 respectively.
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Figure 9 depicts again the resulting payoffs for man 0 as a function of women strategies. In this case men make independent choices choosing one of four possible options with equal probability. Figure 10 show the resulting payoffs as a function of woman strategies. As the figures show, the different scenarios present significant differences on payoff topology and maximum payoff values.
The cooperative situation presents the highest payoff compared with the competitive and the independent ones as shown in figure 8. Figures 9 and 10 show that a better payoff may be obtained in the competitive setup compared to the independent one, but on the other hand, also a much lower payoff for other women strategies may be available.
The independent decision scenario is thus characterized by lowest maxima and less variation on payoffs. Section discussion We have considered the dating market decision problem under the quantum mechanics point of view with the addition of entanglement between players states. Women and men are represented with quantum states and strategies are represented by means of unitary operator on a complex Hilbert space. Men final payoff, considering payoff as a measure of satisfaction, depends on the woman he is paired with.
If men decision states are entangled, their actions are non-locally correlated modeling competition or cooperation scenarios. Three examples are shown in order to illustrate the more usual scenarios.
In two of them the men strategies are correlated in a cooperative and a competitive way respectively. In the other example men strategies are independent. Although cooperative and competitive strategies can drive to higher payoffs, changing of women preferences on those scenarios can lead to very low payoffs. The independent decision scenario is characterized by less variation on payoffs.
Payoff for man 0 if men never choose the same woman, as function of women acceptation strategies. And Note never take an approach too serious. Go out dere and catch fun. And pls don't forget Nothing will make me happier than having a big booty girl as a gift from santa.
What I want for december is a big booty girl. It is no longer news dat "" the one who invests more in a relationship or dating cover or chat, is d one dat is more attracted".
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Do u call her more dan ten times a day and she barely flashes u or sends u pls call me?. Are u d one doing stuffs for her all d time?
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Don't forget attraction is emotional. A simple way of acting like her bf alrdy. Or let's take for example. D basic point is. Sup wit u babe. U want her as ur girl before xmas. Don't ask if she has a bf alrdy. Dat will be going bk to square one. I feel like I av known u. I just feel free and open wit u"" she will most likely reciprocate.
And become open n free wit u. I will explain dese concepts for u.