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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2791Average Radius in the Unit DiskA point is drawn uniformly from the unit disk. What is the expected distance from the point to the center?脑筋急转弯中等derivation未尝试面试订阅2792Right-Hand Circular Segment ProbabilityA point is drawn uniformly from the unit disk. What is the probability that its x-coordinate exceeds 1/2?脑筋急转弯困难derivation未尝试面试订阅2793Expected Chord Length on the Unit CircleTwo points are chosen independently and uniformly on the unit circle. What is the expected length of the chord connecting them?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2795Four Points in a Common SemicircleFour points are chosen independently and uniformly on the unit circle. What is the probability that all four lie inside some semicircle?脑筋急转弯中等derivation未尝试面试订阅2797Expected Squared Distance in the Unit DiskTwo points are chosen independently and uniformly from the unit disk. What is the expected squared Euclidean distance between them?脑筋急转弯中等derivation未尝试面试订阅