A Selfless Person Approaches Jones And Smith With A 100 Bill

A Selfless Person Approaches Jones And Smith With A 100 Bill And Offe

A selfless person approaches Jones and Smith with a $100 bill and offers to sell it to the highest bidder, but the winning and losing bidders must pay her their bids. So if Jones bids $2 and Smith bids $1, they pay a total of $3, and Jones receives the $3, resulting in a net gain of $98 for Jones and a loss of $1 for Smith. If both bid the same amount, the $100 is split evenly between them. Assume that each of them has only two $1 bills on hand, leaving three possible bids: $0, $1, or $2. Write out the payoff matrix for this game and then find its Nash equilibrium.

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The game described involves strategic bidding between two players, Jones and Smith, who each have limited bids ($0, $1, or $2) and aim to maximize their respective payoffs. To analyze this game, it is essential to construct its payoff matrix and determine the Nash equilibrium, which signifies the optimal strategies for both players where neither can improve their outcome by unilaterally changing their bid.

Construction of the Payoff Matrix

We begin by listing all combinations of bids for Jones and Smith and computing their respective payoffs based on the rules:

1. Each bids either $0, $1, or $2.

2. The highest bidder wins the $100 bill, and both players pay their bids.

3. If there is a tie, the $100 is split equally.

The payoff for each player equals their share of the split (or the entire amount if they win outright) minus their bid cost.

Case 1: Jones Bids $0, Smith Bids $0

- Both bid the same; split $100 equally.

- Payoff: Jones = $50, Smith = $50

Case 2: Jones Bids $0, Smith Bids $1

- Smith wins, pays $1, receives $100.

- Jones: pays $0, gets nothing, so net = $0.

- Smith: receives $100, pays $1, net = $99.

Case 3: Jones Bids $0, Smith Bids $2

- Smith wins, pays $2, receives $100.

- Jones: net = $0.

- Smith: net = $98.

Case 4: Jones Bids $1, Smith Bids $0

- Jones wins, pays $1, receives $100.

- Jones: net = $99.

- Smith: net = $0.

Case 5: Jones Bids $1, Smith Bids $1

- Tie: split $100 equally.

- Payoffs: Jones = $50 - $1 = $49, Smith = $50 - $1= $49.

Case 6: Jones Bids $1, Smith Bids $2

- Smith wins, pays $2, net = $98.

- Jones: net = $0.

Case 7: Jones Bids $2, Smith Bids $0

- Jones wins, pays $2, net = $98.

- Smith: net = $0.

Case 8: Jones Bids $2, Smith Bids $1

- Jones wins, pays $2, net = $98.

- Smith: net = $0.

Case 9: Jones Bids $2, Smith Bids $2

- Tie: split $100, each pays $2.

- Payoffs: Jones = $50 - $2 = $48, Smith = $48.

Organizing these into matrices:

| | Smith Bid $0 | Smith Bid $1 | Smith Bid $2 |

|----------|--------------|--------------|--------------|

| Jones Bid $0 | (50,50) | (0,99) | (0,98) |

| Jones Bid $1 | (49,0) | (49,49) | (0,98) |

| Jones Bid $2 | (98,0) | (98,0) | (48,48) |

Analysis to Find Nash Equilibrium

A Nash equilibrium occurs when neither player can improve their payoff by unilaterally changing bids.

Analyzing each cell:

- (Jones $0, Smith $0): Both get $50; unilateral change by either to $1 or $2 improves their payoff? Yes. For Jones, bidding $1$ against Smith's $0$ gives payoff $49$, which is less than $50, so no incentive to deviate. Smith's payoff similar.

- (Jones $0, Smith $1): Smith has $99$, Jones has $0$. Smith could deviate: bidding $2$ to win and get $98$, which is less than $99$. No incentive to deviate.

- (Jones $0, Smith $2): Smith's payoff is $98$; Smith might bid $1$ to tie, split $50$, losing the residual advantage, but since Smith's payoff with bid $2$ is higher, Smith has no incentive to deviate from bidding $2$.

Similarly, check the other cells:

- (Jones $2, Smith $0): Jones has $98$, Smith has $0$. Smith can deviate to $1$, improving payoff to $49$, which is less than current total payoff, unprofitable.

- (Jones $2, Smith $2): payoff is ($48, $48). Deviations would not improve payoff because any unilateral deviation results in worse or equal payoff.

Summarizing, the pure-strategy Nash equilibria are at:

- (Jones bid $0$, Smith bid $0$)

- (Jones bid $2$, Smith bid $2$)

These are the pairs where neither can unilaterally deviate to improve their payoff, given the other’s bid.

Conclusion

The game has two pure-strategy Nash equilibria: both players bidding $0$, and both bidding $2$. From a strategic perspective, these equilibrium points suggest that if both players bid $0$, no one has an incentive to increase bids, resulting in a fair split. Conversely, at bids of $2$, both attempt to secure the full prize, but mutual high bids also keep them at equilibrium since unilateral decreases would result in losing the game without an improvement.

Implications and Strategic Insights

The analysis illustrates that limited bids and strategic considerations significantly influence bidding behavior. When both players bid zero, the outcome is equitable, but such a bid also ensures accounting for the possibility of winning with minimal cost. Conversely, mutual high bidding reveals a contest for dominance, which, although potentially lucrative, carries the risk of overbidding and no guaranteed advantage.

This example reflects broader principles in auction theory and strategic bidding, emphasizing the importance of bid limits and strategic equilibrium considerations in such settings. Understanding these equilibria aids in designing auctions or bidding strategies in real-world scenarios where participants face limited resources or bids.

References

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