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# If there is a majority winner in an election let say candidate $\mathrm{X}$, then candidate $X$ will also be the pairwise sequential winner. then candidate $X$ will also be the plurality method winner. then candidate $\mathrm{X}$ will also be the Condorcet method winner. all the the statements are true. Arrow's Impossibility Theorem implies that it is impossible to have a voting method that satisfies all of the fairness criteria. that in every election, each of the voting methods must produce a different winner. the Plurality Method always produces a majority winner. that there is a voting method that satisfies all of the fairness criteria. An election is held among four candidates (A, B, C, and D). Using a voting method we will call $X$, the winner of the election is candidate A. However, candidate $D$ beats each other candidate in a head to head, pairwise comparison. Based on this information, we can say that voting method $X$ violates the monotonicity criterion. Condorcet criterion. majority criterion. independence of irrelevant alternatives criterion. Consider the game described by the following table. What is the best response for the column player if $s /$ he knows that the row player will make the $\mathrm{Y}$ move? A There is no definitive answer. B C Kodak is contemplating entering (In) the instant photography market and Polaroid can either fight (F) the entry or accommodate (A). The extensive form of this game is shown below: What do you think is the best move for Kodak? F A Out In Consider the following game in normal form. What is the maximin strategy of the Row Player? \begin{tabular}{|c|c|c|c|c|} \hline & & \multicolumn{3}{|c|}{ COLUMN PLAYER } \\ \hline & A & B & $C$ \\ \cline { 2 - 5 } & $X$ & 60,50 & 50,110 & 70,90 \\ \cline { 2 - 5 } ROW & Y & & & \\ \cline { 2 - 5 } & PLYYR & 100,60 & 70,70 & 110,100 \\ \cline { 2 - 5 } & & 110,70 & 100,110 & 40,40 \\ \hline \end{tabular} Z Y $\mathrm{X}$ There is no maximin strategy for the Row Player. Two hunters have set off to get food from the forest. They have two possible game that they can hunt: either a stag, or rabbits. If both players hunt the stag, then they succeed in bringing it down; this gives them both a payoff of 2 units of food. If one player hunts the stag and the other hunts rabbits, then the stag hunter will fail while the rabbit hunter will succeed; i.e. the stag hunter gets a payoff of 0 while the rabbit hunter gets a payoff of 1 . Finally, if both hunters pursue rabbits, they both receive a payoff of 1 . The payoff matrix for this game is shown below. Determine all the Nash equilibria of this game (if there are any). \begin{tabular}{|c|c|c|c|} \hline & \multicolumn{3}{|c|}{ COLUMN PLAYER } \\ \hline \multirow{3}{*}{$\begin{array}{c}\text { ROW } \\ \text { PLAYER }\end{array}$} & & STAG & RABBIT \\ \cline { 2 - 4 } & STAG & 2,2 & 0,1 \\ \cline { 2 - 4 } & RABBIT & 1,0 & 1,1 \\ \hline \end{tabular} (Stag, Stag) and (Rabbit, Rabbit) The game has no Nash equilibrium. (Rabbit, Rabbit) only (Stag, Stag) only Consider the game having the following normal form. Which of the following strategies of the row player is dominated? Z There are no dominated strategies for the row player. Y $x$ Consider the following game in normal form: \begin{tabular}{|c|c|c|c|c|} \hline \multirow{2}{*}{} & \multicolumn{3}{|c|}{ Player B } \\ \cline { 2 - 5 } & $B_{1}$ & $B_{2}$ & $B_{3}$ \\ \hline \multirow{3}{*}{ Player $\mathrm{A}$} & $A_{1}$ & 9,9 & 2,4 & 1,11 \\ \cline { 2 - 5 } & $A_{2}$ & 3,2 & 11,0 & 2,7 \\ \cline { 2 - 5 } & $A_{3}$ & 10,3 & 4,2 & 9,11 \\ \hline \end{tabular} Which of the following is TRUE? The game has dominant equilibrium $A_{3}, B_{3}$. $A_{3}$ is a dominant strategy for Player $\mathrm{A}$. $B_{3}$ is a dominant strategy for Player B. $A_{3}$ dominates $A_{2}$.  