{"id":32203172,"url":"https://github.com/indrag49/qgametheory","last_synced_at":"2026-02-22T19:04:57.098Z","repository":{"id":56936272,"uuid":"264378364","full_name":"indrag49/QGameTheory","owner":"indrag49","description":"General purpose toolbox for simulating quantum versions of game theoretic models. 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Quantum versions of models that have been handled are: Penny Flip Game, Prisoner's Dilemma, Two Person Duel, Battle of the Sexes, Newcomb's Paradox, Hawk and Dove Game, and Monty Hall Problem.\n\nNews: Selected as one of the JUNE 2020: \"Top 40\" New CRAN Packages, picked by Joseph Rickert [@RStudioJoe](https://twitter.com/RStudioJoe)\n\n1. [R Community Blog](https://rviews.rstudio.com/2020/07/27/june-2020-top-40-new-cran-packages/)\n\n2. [R-bloggers](https://www.r-bloggers.com/june-2020-top-40-new-cran-packages/)\n\n---\n\n## Installation\n\nThe development version of the package can be installed from the github repository:\n\n```{r}\ninstall.packages(\"devtools\")\ndevtools::install_github(\"indrag49/QGameTheory\")\n```\nThe released version of **QGameTheory** can be installed from CRAN:\n\n```{r}\ninstall.packages(\"QGameTheory\")\n```\n\n## Dependencies\n\n*QGameTheory* depends on three more packages:\n\n```{r}\nlibrary(dplyr)\nlibrary(RColorBrewer)\nlibrary(R.utils)\n```\n## Global Variables\n\nThe variables that are required for the quantum game theoretic models, are built by initializing:\n\n```{r}\ninit()\n```\nWhere an environement has been developed for holding the variables:\n\n```{r}\nQ \u003c\u003c- new.env(parent=emptyenv())\n```\n\nAll the parameters/variables in the environment are made visible by:\n\n```{r}\nls(Q)\n```\nThe simulator has access to maximum six qubits for quantum computations. Qubits |1\u003e, |0110\u003e and |111110\u003e can be simulated as:\n\n```{r}\nQ$Q1\nQ$Q0110\nQ$Q111110\n```\nThis code chunk when run, produces:\n\n```{r}\n\u003e Q$Q1\n     [,1]\n[1,]    0\n[2,]    1\n\u003e Q$Q0110\n      [,1]\n [1,]    0\n [2,]    0\n [3,]    0\n [4,]    0\n [5,]    0\n [6,]    0\n [7,]    1\n [8,]    0\n [9,]    0\n[10,]    0\n[11,]    0\n[12,]    0\n[13,]    0\n[14,]    0\n[15,]    0\n[16,]    0\n\u003e Q$Q111110\n      [,1]\n [1,]    0\n [2,]    0\n [3,]    0\n [4,]    0\n [5,]    0\n [6,]    0\n [7,]    0\n [8,]    0\n [9,]    0\n[10,]    0\n[11,]    0\n[12,]    0\n[13,]    0\n[14,]    0\n[15,]    0\n[16,]    0\n[17,]    0\n[18,]    0\n[19,]    0\n[20,]    0\n[21,]    0\n[22,]    0\n[23,]    0\n[24,]    0\n[25,]    0\n[26,]    0\n[27,]    0\n[28,]    0\n[29,]    0\n[30,]    0\n[31,]    0\n[32,]    0\n[33,]    0\n[34,]    0\n[35,]    0\n[36,]    0\n[37,]    0\n[38,]    0\n[39,]    0\n[40,]    0\n[41,]    0\n[42,]    0\n[43,]    0\n[44,]    0\n[45,]    0\n[46,]    0\n[47,]    0\n[48,]    0\n[49,]    0\n[50,]    0\n[51,]    0\n[52,]    0\n[53,]    0\n[54,]    0\n[55,]    0\n[56,]    0\n[57,]    0\n[58,]    0\n[59,]    0\n[60,]    0\n[61,]    0\n[62,]    0\n[63,]    1\n[64,]    0\n```\nThe identity matrix:\n\n```{r}\n\u003e Q$I2\n     [,1] [,2]\n[1,]    1    0\n[2,]    0    1\n```\nThe Pauli-X, Pauli-Y and the Pauli-Z matrix:\n\n```{r}\n\u003e sigmaX(Q$I2)\n     [,1] [,2]\n[1,]    0    1\n[2,]    1    0\n\u003e sigmaY(Q$I2)\n     [,1] [,2]\n[1,] 0+0i 0-1i\n[2,] 0+1i 0+0i\n\u003e sigmaZ(Q$I2)\n     [,1] [,2]\n[1,]    1    0\n[2,]    0   -1\n```\n\nThe Hadamard Gate:\n\n```{r}\n\u003e Hadamard(Q$I2)\n          [,1]       [,2]\n[1,] 0.7071068  0.7071068\n[2,] 0.7071068 -0.7071068\n```\nThe application of Pauli-X gate on |0\u003e and on |1\u003e, i.e, the spin flip operations on qubits, can be simulated in the following way:\n\n```{r}\n\u003e sigmaX(Q$Q0)\n     [,1]\n[1,]    0\n[2,]    1\n\u003e sigmaX(Q$Q1)\n     [,1]\n[1,]    1\n[2,]    0\n```\nThere are other important quantum gates like: CNOT, Fredkin, Toffoli, T, Phase, Rx, etc.\n\n```{r}\n\u003e CNOT(Q$I4)\n     [,1] [,2] [,3] [,4]\n[1,]    1    0    0    0\n[2,]    0    1    0    0\n[3,]    0    0    0    1\n[4,]    0    0    1    0\n\u003e CNOT(Q$Q11)\n     [,1]\n[1,]    0\n[2,]    0\n[3,]    1\n[4,]    0\n\u003e Fredkin(Q$Q110)\n     [,1]\n[1,]    0\n[2,]    0\n[3,]    0\n[4,]    0\n[5,]    0\n[6,]    1\n[7,]    0\n[8,]    0\n\u003e Toffoli(Q$Q010)\n     [,1]\n[1,]    0\n[2,]    0\n[3,]    1\n[4,]    0\n[5,]    0\n[6,]    0\n[7,]    0\n[8,]    0\n\u003e T(Q$Q_minus)\n                [,1]\n[1,]  0.7071068+0.0i\n[2,] -0.5000000-0.5i\n\u003e Phase(Q$I2)\n     [,1] [,2]\n[1,] 1+0i 0+0i\n[2,] 0+0i 0+1i\n\u003e Phase(Q$Q_plus)\n                     [,1]\n[1,] 0.7071068+0.0000000i\n[2,] 0.0000000+0.7071068i\n\u003e Rx(Q$Q1, pi/3)\n                    [,1]\n[1,] 0.9659258+0.000000i\n[2,] 0.0000000-0.258819i\n```\nOne can prepare one of the 4 Bell states by using:\n\n```{r}\n\u003e Bell(Q$Q0, Q$Q1)\n          [,1]\n[1,] 0.0000000\n[2,] 0.7071068\n[3,] 0.7071068\n[4,] 0.0000000\n\u003e Bell(Q$Q1, Q$Q1)\n           [,1]\n[1,]  0.0000000\n[2,]  0.7071068\n[3,] -0.7071068\n[4,]  0.0000000\n```\n\nThe **Quantum Fourier Transform** for a given state |y\u003e is simulated by:\n\n```{r}\n\u003e QFT(4)\n              [,1]\n[1,]  0.3535534+0i\n[2,] -0.3535534+0i\n[3,]  0.3535534-0i\n[4,] -0.3535534+0i\n[5,]  0.3535534-0i\n[6,] -0.3535534+0i\n[7,]  0.3535534-0i\n[8,] -0.3535534+0i\n```\n\nFinally for preparing and measuring an arbitrary quantum state,\n\n```{r}\n\u003e sigma_x \u003c- sigmaX(Q$I2)\n\u003e U \u003c- (kronecker(Q$I2, Q$I2)+1i*kronecker(sigma_x, sigma_x))/sqrt(2)\n\u003e Psi \u003c- U %*% Q$Q00\n\u003e Psi\n                     [,1]\n[1,] 0.7071068+0.0000000i\n[2,] 0.0000000+0.0000000i\n[3,] 0.0000000+0.0000000i\n[4,] 0.0000000+0.7071068i\n\u003e QMeasure(Psi)\n```\n\n\u003cimg src=\"man/figures/1.png\" alt=\"\"/\u003e\n\n## Game Theory Concepts\n\nThe **Iterated Deletion of Strictly Dominated Strategies** algorithm is simulated by taking in the payoff matrices of two players from a two-person game:\n\n```{r}\n\u003e P1 \u003c- matrix(c(8, 0, 3, 3, 2, 4, 2, 1, 3), ncol=3, byrow=TRUE)\n\u003e P2 \u003c- matrix(c(6, 9, 8, 2, 1, 3, 8, 5, 1), ncol=3, byrow=TRUE)\n\u003e IDSDS(P1, P2)\n[[1]]\n     [,1]\n[1,]    4\n\n[[2]]\n     [,1]\n[1,]    3\n```\n\nThe **NASH** equilibrium of the payoff matrix of a two person game is computed similary in the following way:\n\n```{r}\n\u003e NASH(P1, P2)\nJoining, by = c(\"V1\", \"V2\")\n  V1 V2\n1  2  3\n```\nThis generates the indices of the cell corresponding to the NASH equilibrium.\n\n## Quantum Game Theory Models\n\n### Quantum Penny Flip\n\nFor the game tree mentioned below:\n\n\u003cimg src=\"man/figures/2.png\" alt=\"\"/\u003e\n\nThe simulation codes go in the following way:\n\n```{r}\n\u003e psi \u003c- (Q$Q0+Q$Q1)/sqrt(2)\n\u003e S1 \u003c- sigmaX(Q$I2)\n\u003e S2 \u003c- Q$I2\n\u003e H \u003c- Hadamard(Q$I2)\n\u003e SA \u003c- list(S1, S2)\n\u003e SB \u003c- list(H)\n\u003e QPennyFlip(psi,SA,SB)\n```\nIt produces the plot:\n\n\u003cimg src=\"man/figures/3.png\" alt=\"\"/\u003e\n\n### Quantum Prisoner's Dilemma\n\nOne instance of the Quantum Prisoner's Dilemma game can be simulated first by providing the strategies played by both Alice and Bob along with the payoffs *w, x, y, z* available to them corresponding to their choices. The payoffs follow, *z\u003ew\u003ex\u003ey*. \n\n```{r}\n\u003e QPD(Hadamard(Q$I2), sigmaZ(Q$I2), 3, 1, 0, 5)\n[1] 1.5 4.0\n```\n\nIt also generates the plot:\n\n\u003cimg src=\"man/figures/4.png\" alt=\"\"/\u003e\n\nThe payoff matrix of the **Quantum Prisoner's Dilemma** for both the players can be constructed:\n\n```{r}\n\u003e moves \u003c- list(Q$I2, sigmaX(Q$I2), Hadamard(Q$I2), sigmaZ(Q$I2))\n\u003e PayoffMatrix_QPD(moves, 3, 1, 0, 5)\n[[1]]\n     [,1] [,2] [,3] [,4]\n[1,]    3    0 0.50  1.0\n[2,]    5    1 0.50  0.0\n[3,]    3    3 2.25  1.5\n[4,]    1    5 4.00  3.0\n\n[[2]]\n     [,1] [,2] [,3] [,4]\n[1,]  3.0  5.0 3.00    1\n[2,]  0.0  1.0 3.00    5\n[3,]  0.5  0.5 2.25    4\n[4,]  1.0  0.0 1.50    3\n```\n\nThe above code also generates all the sixteen possible combinations of plots. Analysing, it is noticed that the quantum version helps us escape the so called dilemma in the classical Prisoner's dilemma game. One next uses the **IDSDS** algorithm to find the **strictly dominant strategy equilibrium**. The **NASH** equilibrium is also calculated by the above codes. It can be seen that both of them give the same result and the equilibrium is Pareto Optimum too.\n\n### Quantum Two Person Duel\n\nSimulation is carried out to calculate the expected payoffs to Alice and Bob for the following three cases:\n\n1) The game is continued for 'n' rounds and none of the players shoots at the air.\n\n2) The game is continued for 2 rounds and Alice shoots at the air in her second round.\n\n3) The game is continued for 2 rounds and Bob shoots at the air in her second round.\n\n```{r}\n\u003e Qs \u003c- (Q$Q0+Q$Q1)/sqrt(2)\n\u003e Psi \u003c- kronecker(Qs, Qs)\n\u003e QDuels_Alice_payoffs(Psi, 5, 0.666666, 0.5, 0, 0, 0.2, 0.7)\n[1] 0.2087876 0.3281732 0.4894636\n\u003e QDuels_Bob_payoffs(Psi, 5, 0.666666, 0.5, 0, 0, 0.2, 0.7)\n[1] 0.7912124 0.6718268 0.5105364\n```\n\nFour plotting functions are available to simulate the corresponding results:\n\n1) For plotting Alice's and Bob's expected payoffs as functions of 'alpha1' and 'alpha2':\n\n```{r}\nQDuelsPlot1(Psi, 5, 0.66666, 0.5, 0.2, 0.7)\n```\n\n\u003cp float=\"left\"\u003e\n  \u003cimg src=\"man/figures/5.png\" /\u003e\n  \u003cimg src=\"man/figures/6.png\" /\u003e \n\u003c/p\u003e\n\n2) For plotting Alice's and Bob's expected payoffs as functions of the number of rounds 'n' played in a repeated quantum duel\n\n```{r}\nQDuelsPlot2(Psi, 5, 0.666666, 0.5, 0, 0, 0.2, 0.7)\n```\n\u003cp float=\"left\"\u003e\n  \u003cimg src=\"man/figures/7.png\" /\u003e\n  \u003cimg src=\"man/figures/8.png\" /\u003e \n\u003c/p\u003e\n\n3) For plotting the improvement in Alice's expected payoff as a function of 'a' and 'b', if Alice chooses to fire at the air in her second shot, in a two round game\n\n```{r}\nQDuelsPlot3(Psi, 0, 0)\n```\n\n\u003cimg src=\"man/figures/9.png\" alt=\"\"/\u003e\n\n4) For plotting the improvement in Bob's expected payoff as a function of 'a' and 'b', if Bob chooses to fire at the air in her second shot, in a two round game\n\n```{r}\nQDuelsPlot4(Psi, 0, 0)\n```\n\n\u003cimg src=\"man/figures/10.png\" alt=\"\"/\u003e\n\n### Quantum Battle of the Sexes\n\nOne instance for the **Quantum Battle of the Sexes** can be computed for a particular set of probability values:\n\n```{r}\n\u003e moves \u003c- list(Q$I2, sigmaX(Q$I2))\n\u003e QBOS(0, 1, moves, 5, 3, 1)\n[1] 1.875 2.375\n```\nThe payoff matrix for the quantum game consisting of all the possible combinations of the probabilities can also be constructed:\n\n```{r}\n\u003e PayoffMatrix_QBOS(moves, 5, 3, 1)\n[[1]]\n      [,1]  [,2]\n[1,] 2.875 1.875\n[2,] 2.375 2.875\n\n[[2]]\n      [,1]  [,2]\n[1,] 2.875 2.375\n[2,] 1.875 2.875\n```\n\n### Quantum Newcomb's Paradox\n\nThe quantum version of the Newcomb's Paradox can be simulated by taking in the choice of the qubit |0\u003e or |1\u003e by the supercomputer 'Omega' and the probability with which Alice plays the spin flip operator as the input parameters.\n\n```{r}\n\u003e QNewcomb(Q$Q1, 0)\n     [,1]\n[1,]    0\n[2,]    0\n[3,]    0\n[4,]    1\n```\n\u003cimg src=\"man/figures/11.png\" alt=\"\"/\u003e\n\n### Quantum Hawk and Dove\n\nOne instance of the **Quantum Hawk and Dove** game can be simulated for a particular set of probability values:\n\n```{r}\n\u003e moves \u003c- list(Q$I2, sigmaX(Q$I2))\n\u003e QHawkDove(0, 1, moves, 50, -100, -10)\n[1] 18.75 18.75\n```\n\nThe payoff matrix for the quantum game consisting of all the possible combinations of the probabilities can also be constructed:\n\n```{r}\n\u003e PayoffMatrix_QHawkDove(moves, 50, -100, -10)\n[[1]]\n         [,1]    [,2]\n[1,] 15.78125 21.9375\n[2,] 24.53125 28.4375\n\n[[2]]\n         [,1]    [,2]\n[1,] 28.28125 21.9375\n[2,] 24.53125 15.9375\n```\n\n### Quantum Monty Hall Problem\n\nThe qutrits required for this problem can be constructed in the following way:\n\n```{r}\n\u003e Q$Qt0\n     [,1]\n[1,]    1\n[2,]    0\n[3,]    0\n\u003e Q$Qt1\n     [,1]\n[1,]    0\n[2,]    1\n[3,]    0\n\u003e Q$Qt2\n     [,1]\n[1,]    0\n[2,]    0\n[3,]    1\n```\n\nSome SU(3) matrices:\n\n```{r}\n\u003e Q$Identity3\n     [,1] [,2] [,3]\n[1,]    1    0    0\n[2,]    0    1    0\n[3,]    0    0    1\n\u003e Q$Hhat\n                      [,1]                  [,2]                 [,3]\n[1,]  0.7071068+0.0000000i  0.5000000+0.0000000i 0.5000000+0.0000000i\n[2,] -0.5000000+0.0000000i  0.5303301-0.4677072i 0.1767767+0.4677072i\n[3,] -0.1767767-0.4677072i -0.3750000+0.3307189i 0.6250000+0.3307189i\n```\n\nA quantum state is prepared first from the qutrits:\n\n```{r}\n\u003e Psi_in \u003c- kronecker(Q$Qt0, (Q$Qt00+Q$Qt11+Q$Qt22)/sqrt(3))\n\u003e Psi\n     [,1]\n[1,]  0.5\n[2,]  0.5\n[3,]  0.5\n[4,]  0.5\n```\n\nThe **Quantum Monty Hall** problem, next, is simulated in the following way:\n\n```{r}\n\u003e QMontyHall(Psi_in, pi/4, Q$Identity3, Q$Hhat)\n[1] 0.125 0.875\n```\n\nIt returns the expected payoffs to Alice and Bob after the end of the game.\n\n## Miscellaneous functions\n\nSome of the functions that are required for the analyses are:\n\n### row_count()\n\nThe above function calculates the number of rows in a matrix or a vector\n\n```{r}\n\u003e row_count(Q$Q01)\n[1] 4\n\u003e row_count(Q$I8)\n[1] 8\n```\n\n### col_count()\n\nThe above function calculates the number of columns in a matrix or a vector\n\n```{r}\n\u003e col_count(Q$Q01)\n[1] 1\n\u003e col_count(Q$I8)\n[1] 8\n```\n\n### levi_civita()\n\nCalculates the Levi-Civita function for the integers: 0, 1 and 2\n\n```{r}\n\u003e levi_civita(0, 2, 1)\n[1] -1\n\u003e levi_civita(1, 2, 0)\n[1] 1\n\u003e levi_civita(1, 2, 1)\n[1] 0\n```\n\n---\n# Some Useful References\n1. https://arxiv.org/abs/quant-ph/0208069\n\n2. https://arxiv.org/abs/quant-ph/9804010\n\n3. https://arxiv.org/abs/quant-ph/0506219\n\n4. https://arxiv.org/abs/quant-ph/0305058\n\n5. https://arxiv.org/abs/quant-ph/0110096\n\n6. https://arxiv.org/abs/quant-ph/0108075\n\n7. https://arxiv.org/abs/quant-ph/0202074\n\n8. https://arxiv.org/abs/quant-ph/0109035\n\n9. https://github.com/corbett/QuantumComputing\n\n10. https://github.com/tvganesh/QCSimulator\n\n11. https://github.com/indrag49/Quantum-SimuPy\n\n12. https://www.amazon.in/Quantum-Computation-Information-10th-Anniversary/dp/1107002176\n\n13. https://arxiv.org/abs/1512.06808\n---\n\nLogo designed by Manash Kashyap\n","project_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Findrag49%2Fqgametheory","html_url":"https://awesome.ecosyste.ms/projects/github.com%2Findrag49%2Fqgametheory","lists_url":"https://awesome.ecosyste.ms/api/v1/projects/github.com%2Findrag49%2Fqgametheory/lists"}