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2766Matching PenniesIn the zero-sum game of matching pennies, Row and Column each choose H or T. Row receives +1 if the choices match and -1 otherwise. Find the mixed-strategy equilibrium and the value of the game.脑筋急转弯简单derivation未尝试面试订阅2767Weighted Matching PenniesConsider the zero-sum matrix game \[ \begin pmatrix 2 & -1 \\ -1 & 2 \end pmatrix . \] Find the optimal mixed strategies and the value of the game.脑筋急转弯中等derivation未尝试面试订阅2768Penalty Kick as a Zero-Sum GameA striker chooses Left or Right, and a goalkeeper chooses Left or Right. The striker's success probabilities are \[ \begin pmatrix 0.6 & 0.9 \\ 0.8 & 0.7 \end pmatrix , \] where rows are the striker's choices and columns are the keeper's choices. Treat this as a zero-sum game with Row = striker. Find the equilibrium mixing probabilities and the value.脑筋急转弯中等derivation未尝试面试订阅2769General 2x2 Zero-Sum FormulaFor the zero-sum matrix game \[ \begin pmatrix a & b \\ c & d \end pmatrix , \] assume there is no pure saddle point and a-b-c+d 0. Derive the mixed strategy of Row, the mixed strategy of Column, and the value of the game.脑筋急转弯困难derivation未尝试面试订阅2770Spotting a Saddle PointConsider the zero-sum matrix \[ \begin pmatrix 4 & 1 & 3 \\ 2 & 2 & 2 \\ 5 & 0 & 4 \end pmatrix . \] Does the game have a saddle point in pure strategies? If so, identify it and give the value.脑筋急转弯简单derivation未尝试面试订阅2771Searching Boxes With Unequal StakesA searcher chooses one of three boxes to inspect; a hider chooses one box to hide in. If the searcher picks the correct box, the searcher earns the box's value; otherwise the payoff is 0. The three box values are 6, 3, and 2. Find the searcher's optimal mixed strategy and the value of the game.脑筋急转弯中等derivation未尝试面试订阅2772Patrolling Two TargetsA defender can patrol either Target 1 or Target 2. An attacker chooses which target to strike. If the defender patrols the attacked target, damage is prevented and the defender's payoff is 0. If not, the defender loses the target's damage: 4 for Target 1 and 1 for Target 2. Find the defender's optimal mixed strategy and the value of the game.脑筋急转弯中等derivation未尝试面试订阅2773Continuous Defense SplitA defender allocates a fraction x of one unit of defense budget to Target 1 and the remaining 1-x to Target 2. If the attacker strikes Target 1, the defender's payoff is -9(1-x); if the attacker strikes Target 2, the defender's payoff is -6x. Find the defender's optimal allocation and the value.脑筋急转弯中等derivation未尝试面试订阅2774Weighted Rock-Paper-ScissorsConsider the zero-sum matrix \[ \begin pmatrix 0 & -1 & 2 \\ 1 & 0 & -1 \\ -2 & 1 & 0 \end pmatrix . \] Find the optimal mixed strategy and the value of the game.脑筋急转弯中等derivation未尝试面试订阅2775Adding a Constant to Every EntrySuppose A is a zero-sum payoff matrix with value v. If we add the same constant c to every entry of A, what happens to the optimal mixed strategies and to the value? Explain briefly.脑筋急转弯简单derivation未尝试面试订阅2776Scaling the Whole Payoff MatrixSuppose A is a zero-sum payoff matrix with value v, and >0. What happens to the optimal mixed strategies and the game value if the payoff matrix becomes A?脑筋急转弯简单derivation未尝试面试订阅2777Reduce a 3x3 Game Before SolvingConsider the zero-sum matrix \[ \begin pmatrix 3 & 0 & 4 \\ 2 & -1 & 2 \\ 1 & 2 & 3 \end pmatrix . \] Identify any dominated strategy that can be removed, reduce the game, and then solve for the mixed equilibrium and value.脑筋急转弯困难derivation未尝试面试订阅2778Overlapping Search PatternsA searcher can use one of two search patterns: Pattern 1 checks locations A and B, while Pattern 2 checks locations B and C. The hider chooses one location. Row's payoff is 1 if the chosen pattern covers the hider's location and 0 otherwise. Find the equilibrium and the value.脑筋急转弯中等derivation未尝试面试订阅2779Imperfect Defense on Two RoutesA defender chooses whether to patrol Route L or Route R. An attacker chooses which route to use. The defender's payoff matrix is \[ \begin pmatrix -0.2 & -2.0 \\ -1.0 & -0.3 \end pmatrix , \] where rows are the defender's choices and columns are the attacker's choices. Find the equilibrium mixes and the value to the defender.脑筋急转弯困难derivation未尝试面试订阅2780Why Supported Pure Actions Must TieExplain why, in any mixed-strategy equilibrium of a finite zero-sum game, every pure strategy used with positive probability by a player must yield the same expected payoff against the opponent's equilibrium mix.脑筋急转弯中等derivation未尝试面试订阅2781Why Strictly Dominated Strategies Can Be RemovedIn a finite zero-sum game, why does removing a strictly dominated pure strategy never change the value of the game? Give a concise justification.脑筋急转弯中等derivation未尝试面试订阅2782A 2x3 Matrix With One Nonbinding ColumnSolve the zero-sum game \[ \begin pmatrix 1 & 0 & 2 \\ 0 & 2 & 1 \end pmatrix . \] Find the optimal mixed strategies and the value.脑筋急转弯中等derivation未尝试面试订阅2783Choosing Between a Fragile and a Robust HedgeA trader chooses between Hedge A and Hedge B. Nature chooses Stress 1 or Stress 2. The trader's PnL matrix is \[ \begin pmatrix 3 & -2 \\ 0 & 1 \end pmatrix . \] Treat Nature as an adversary in a zero-sum game. Find the trader's optimal mix and the value.脑筋急转弯中等derivation未尝试面试订阅2784LP View of a Rectangular Zero-Sum GameConsider the zero-sum game \[ \begin pmatrix 2 & -1 & 0 \\ 0 & 1 & 3 \end pmatrix . \] Write the row player's maximin problem as a linear program, and solve for the optimal mix and the value.脑筋急转弯困难derivation未尝试面试订阅2785High-Penalty Coordination FailureSolve the zero-sum game \[ \begin pmatrix 1 & -3 \\ -3 & 1 \end pmatrix . \] Find the optimal mixed strategies and the value.脑筋急转弯中等derivation未尝试面试订阅