Question

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?

Short answer

Explain briefly. Answer: No, there is no unique Nash Equilibrium in this game. Since each player's strategy depends on the other player's strategy, attempting to find the optimal bid amount would result in an infinite process of outbidding each other. The game becomes even more uncertain if players do not know each other's valuations, making it challenging to accurately predict optimal strategies.

Step-by-step solution

Identify the players, actions, and payoffs

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.

Categorize strategies in the game

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.

Evaluate dominant strategies

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.

Determine Nash Equilibrium

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.

Assess the impact of lack of information

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.

Key concepts

Sealed-bid auction

Sealed-bid auctions are a common method used in auctions where bids are submitted secretly. This means each participant does not know what others have bid. In the context of our baseball card auction, this means player A and player B will make their bids independently, with no knowledge of the amount the other person might offer.

Once all bids have been submitted, the auctioneer opens and compares them. The highest bid wins the auction. If there's a tie, as with equal bids, a coin flip determines the winner.

• Sealed-bid nature prevents strategic behavior based on others' bids.
• Each player must determine their bid based on their valuation alone.

This format leads to interesting strategic decisions because it conceals intentions and anticipated actions, impacting each player's strategy and outcome.

Nash equilibrium

The Nash equilibrium is a key concept in auction theory and game theory. It describes a scenario where each player has chosen a strategy, and no player can benefit by changing their strategy while the others keep theirs unchanged. In simpler terms, it's where everyone's tactics are balanced.

For our auction, finding a Nash equilibrium entails identifying a bid for each player such that neither player would want to adjust their bid, given knowledge of the other's strategy.

• Players are trying to outguess each other.
• Ideal equilibrium would mean neither player sees a benefit in changing their bidding strategy.

However, because players are constantly trying to outbid each other by small amounts, finding a stable Nash equilibrium in a sealed-bid auction can be exceedingly complex or even unattainable.

Dominant strategy

In the context of game theory, a dominant strategy is when one strategy clearly outperforms others, regardless of what the opponent does. It's seen as the safest choice for a player, assuring a better or at least similar outcome as any other strategy.

Unfortunately, in many sealed-bid auctions, identifying a dominant strategy is challenging. In our baseball card auction, there is no single bid amount for either player A or B that guarantees the highest payoff in all scenarios.

• Players rely on anticipated strategies of others.
• Without knowing other bids, a clear dominant strategy is elusive.

Every move depends heavily on predictions, making strategic decisions primarily based on individual valuations and educated guesses rather than guaranteed wins.

Valuation uncertainty

Valuation uncertainty plays a crucial role in auctions, particularly in cases where bidders do not know each other's valuations, such as a common value auction. This uncertainty drives the strategic guessing and adjustments in bidding.

In our scenario, if players A and B did not know each other's valuations, they would have to estimate what the other values the card at and bid accordingly.

• Lack of information increases the game's complexity.
• Players might bid too conservatively or aggressively based on wrong assumptions.

Decisions made under valuation uncertainty often result in inefficiencies, as bidders try to balance their own valuation with guesses about what others might pay, leading to less predictable outcomes.