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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3466Capacity of a Ternary Test With Uneven Follow-UpsA first test has three outcomes. If outcome A occurs, you may ask two binary follow-up questions; if outcome B occurs, you may ask one ternary follow-up; if outcome C occurs, the process must stop. What is the maximum number of equally likely states you can distinguish?数学简单derivation未尝试面试订阅3467Capacity of a Binary Gate With Different Menus on Each SideA first yes/no question is asked. On the yes branch, you may run one four-way diagnostic. On the no branch, you may run two further yes/no diagnostics. What is the maximum number of states that can be distinguished?数学简单derivation未尝试面试订阅3468Capacity of a Four-Way Triage With Partial Follow-UpA first test has four outcomes. On the first two branches you may ask one binary follow-up each; on the last two branches you must stop immediately. What is the maximum number of states you can distinguish?数学简单derivation未尝试面试订阅3469Capacity of a Ternary Gate With a Five-Way EscalationA first test has outcomes left, middle, right. If left occurs, you may run one five-way escalation test. If middle or right occurs, you may run one binary follow-up on that branch. How many states can be distinguished in total?数学简单derivation未尝试面试订阅3470Capacity of a Binary Gate With Deep and Shallow BranchesA first yes/no question is asked. If yes, you may then ask one ternary question followed by one binary question. If no, you may ask only one ternary question. What is the maximum number of distinguishable states?数学简单derivation未尝试面试订阅3471One Ternary Plus Three Binary Assays for One Fault or NoneYou may run one ternary assay and three binary assays non-adaptively. There is either no faulty component or exactly one faulty component. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3472Two Ternary and One Binary Assays for One Fault or NoneYou may run two ternary assays and one binary assay non-adaptively. There is either no faulty component or exactly one faulty component. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3473One Four-Way and Two Binary Assays for One Fault or NoneYou may run one four-way assay and two binary assays non-adaptively. There is either no faulty component or exactly one faulty component. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅3474Binary Assays When Each Faulty Unit Can Fail in Two ModesYou have three yes/no assays run non-adaptively. Either no component is faulty, or exactly one component is faulty and it can be in one of two distinct failure modes. What is the largest number of components you can identify?数学简单derivation未尝试面试订阅