There are three players and a single indivisible object. Each player has a value for the object. All players know the values of other players. Values are as follows player simultaneously and independently submits bid (i.e., bids must be integers) for the

There are three players and a single indivisible object. Each player $i$ has a value $v_{i}$ for the object. All players know the values of other players. Values are as follows

$$v_{1}=7, \quad v_{2}=4, \quad v_{3}=1$$

\nEach player simultaneously and independently submits bid $b_{i} \in\{0,1,2, \ldots\}$ (i.e., bids must be integers) for the object. The winner of the object always has to pay the amount of its own bid while losers pay nothing, i.e., player $i$ gets $v_{i}-b_{i}$ when she wins the object, and 0 otherwise. The exact rules for determining the winner of the object will be specified later. (a) Suppose that the game rules for determining the winner are fair and symmetric. In other words, if all players bid the same amount, the object is awarded randomly to one of them with fair odds. For each player, identify all weakly dominated strategies in such games. (b) Suppose that the winner of the object is the player with the highest bid. Ties (if any) are broken randomly using a fair randomizing device.\nFind all pure-strategy Nash equilibria of this game in which players do not use weakly dominated strategies identified in part (a). For each Nash equilibrium you find, state whether the outcome is socially efficient (maximizes the social welfare). (c) Now suppose that the winner of the object is the player with the secondhighest bid. Ties (if any) are broken randomly using a fair randomizing device. For example, if $b_{1}=3, b_{2}=1, b_{3}=2$, then the winner is Player 3. If $b_{1}=1, b_{2}=1, b_{3}=2$, then the winner is either Player 1 (50% chance) or Player 2 (50% chance). \nFind all pure-strategy Nash equilibria of this game in which players do not use weakly dominated strategies identified in part (a). For each Nash equilibrium you find, state whether the outcome is socially efficient (maximizes the social welfare).