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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5677Three-Pile Nim Winning MoveTwo players play standard Nim with three piles of sizes 3, 5, and 7. On a turn a player removes any positive number of stones from a single pile, and the player who takes the last stone wins. Does the first player win, and if so, give a winning first move (which pile and how many stones to remove).脑筋急转弯简单数值题未尝试免费5678Six Pirates and 100 CoinsSix pirates ranked 1 (most senior) to 6 must divide 100 gold coins. Starting with the most senior, the proposer suggests an allocation; all living pirates (including the proposer) vote. If at least half vote yes the plan passes; otherwise the proposer is thrown overboard and the next most senior proposes. Pirates are perfectly rational and value, in order: survival, then maximizing their own coins, then (as a tiebreaker) seeing others thrown overboard. How many coins does the most senior pirate keep?脑筋急转弯中等数值题未尝试免费5679Race to the Last of 100A single pile holds 100 stones. Players alternate removing between 1 and 7 stones (inclusive). The player who takes the last stone wins. Does the first player win, and if so how many stones should they remove on the first move?脑筋急转弯简单数值题未尝试免费5680First to Say Twenty-OneTwo players build a running total starting at 0. On each turn a player adds 1, 2, or 3 to the total. The player who makes the total reach exactly 21 wins. Does the first player win, and if so what number should they add on the very first move?脑筋急转弯简单数值题未尝试免费5681Misere Nim of SingletonsFive piles each contain exactly 1 stone. Players alternate removing one or more stones from a single pile; here the player forced to take the LAST stone LOSES (misere play). Under optimal play, does the first player win or lose?脑筋急转弯简单brainteaser未尝试免费5682Queen on the Quarter-BoardTwo piles hold 3 and 5 tokens. On a turn a player either removes any positive number from one pile, or removes the SAME positive number from both piles. The player taking the last token (emptying both piles) wins. With the position at (3, 5), does the player to move win or lose under optimal play?脑筋急转弯中等brainteaser未尝试免费