Consider the following sealed-bid auction for a rare baseball card. Player \(A\) values the card being auctioned at \(\$ 600,\) player lvalues the card at \(\$ 500,\) and these valuations are known to each player who will submit a sealed bid for the card. Whoever bids the most will win the card. If equal bids are submitted, the auctioneer will flip a coin to decide the winner. Each player must now decide how much to bid. a. How would you categorize the strategies in this game? Do some strategies dominate others? b. Does this game have a Nash equilibrium? Is it unique? c. How would this game change if each player did not know the other's valuation for the card?

In this game, the players are A and B. Their actions are the bids they submit, which can be any amount between $0 and their own valuation for the rarer baseball card. The payoff for each player is the difference between their valuation of the card and their bid if they win, and 0 if they lose. The winner is the one who bids the most, and in the case of equal bids, the winner is decided by a coin flip.

The strategies for each player can be defined as the bids they submit to the auctioneer. Since the bids can be any amount between $0 and their own valuation, the strategies can be considered continuous strategies. To find if any strategy dominates the others, we have to check if for one player, there is a strategy that gives a higher payoff for each strategy of the other player.

In this game, it is difficult to find a dominant strategy because no strategy always leads to a higher payoff. It depends on the strategy (bid amount) chosen by the other player, which determines whether a strategy gives a higher payoff or not.

A Nash Equilibrium in this game would be a pair of strategies (bids) where neither player would want to deviate from their strategy, given that the other player sticks to their strategy. We can find the Nash Equilibrium by analyzing player A's best response to player B's strategy and vice versa. If player A knows that player B will bid \(x\) and wants to win with the highest payoff, player A will bid \(x+\epsilon\) (where \(\epsilon\) is a very small positive value) to outbid player B while minimizing their own bid. The same logic applies to player B. However, this logic will lead to an infinite process of outbidding each other, which doesn't yield a possible Nash Equilibrium. Therefore, the game does not have a unique Nash Equilibrium in this case.

If the players did not know each other's valuations for the card, the game would become more complex. The players would need to estimate each other's valuation and make decisions based on those estimates. This would lead to a more uncertain and challenging game with no definitive Nash Equilibrium, since players would not have enough information to make accurate predictions about each other's strategies and decide on an optimal bid amount.