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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2786Uniform Mixing in a 3x3 Guessing GameConsider the zero-sum matrix with payoff +1 on the diagonal and -1 off the diagonal: \[ \begin pmatrix 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end pmatrix . \] Find the mixed equilibrium and the value.脑筋急转弯中等derivation未尝试面试订阅2787Why Pure Saddle Points Already Solve the Mixed GameSuppose a finite zero-sum matrix game has a saddle point at entry (i \*,j \*). Explain why allowing mixed strategies cannot improve either player's outcome beyond that same value.脑筋急转弯中等derivation未尝试面试订阅2788Search Three Targets With Values 5, 3, and 2A defender can inspect exactly one of three targets. If the attacker chooses the inspected target, the defender earns the target's value; otherwise the defender gets 0. The target values are 5, 3, and 2. Find the defender's optimal inspection mix and the value.脑筋急转弯中等derivation未尝试面试订阅2789Duplicate Rows Do Not MatterSuppose a zero-sum matrix has two identical rows. Explain why deleting one of the duplicate rows cannot change the value of the game.脑筋急转弯简单derivation未尝试面试订阅2790A Safe Strategy Creates Multiple EquilibriaConsider the zero-sum game \[ \begin pmatrix 1 & -1 \\ 0 & 0 \\ -1 & 1 \end pmatrix . \] Find the value of the game and describe at least one optimal strategy for each player.脑筋急转弯中等derivation未尝试面试订阅2794When a Midpoint-Generated Chord Exceeds the RadiusA chord of the unit circle is generated by choosing its midpoint uniformly in the disk. What is the probability that the chord length exceeds 1, the circle's radius?脑筋急转弯中等derivation未尝试面试订阅2799A Unit-Interval Distance WindowIf U and V are independent Uniform (0,1) variables, what is the probability that |U-V|\le 1/4? Solve it geometrically.脑筋急转弯中等derivation未尝试面试订阅2801Staying Away From All Three SidesA point is chosen uniformly inside an equilateral triangle. What is the probability that its distance to each of the three sides is at least one-sixth of the altitude?脑筋急转弯中等derivation未尝试面试订阅2804Chord Longer Than √2Fix one point on the unit circle and choose a second point uniformly on the circle. What is the probability that the chord length exceeds \sqrt2?脑筋急转弯中等derivation未尝试面试订阅2805Squared Distance to the Center of the SquareA point is chosen uniformly in the unit square. What is the expected squared distance from the point to the center (1/2,1/2)?脑筋急转弯中等derivation未尝试面试订阅2806Closer to the Center Than to the BoundaryA point is chosen uniformly in the unit disk. What is the probability that the point is closer to the center than to the boundary of the disk?脑筋急转弯中等derivation未尝试面试订阅2809Maximum Coordinate of Two Uniform NumbersTwo independent Uniform (0,1) variables U and V are drawn. What is E[\max(U,V)]? Give a geometric interpretation in the unit square.脑筋急转弯中等derivation未尝试面试订阅2810Expected Squared Chord LengthTwo points are chosen independently and uniformly on the unit circle. What is the expected squared length of the chord joining them?脑筋急转弯中等derivation未尝试面试订阅