Give A Formal Proof That Truthfulness In The Second-Price Au ✓ Solved

Give a formal proof that truthfulness in the second-price auction for n players is a dominant strategy

This assignment requires providing a formal proof demonstrating that bidding truthfully (bidding one's true valuation) is a dominant strategy in a second-price auction involving n players. Additionally, the task involves analyzing Bayesian-Nash equilibria in a first-price auction where players’ values are independently drawn from a uniform distribution on [a, b], and computing optimal bids for specific parameters. It also includes examining a specific case with three players and a known valuation for one player, assessing whether deviation improves expected utility.

Sample Paper For Above instruction

Formal Proof of Truthfulness as a Dominant Strategy in Second-Price Auctions

The second-price auction, also known as the Vickrey auction, is a well-studied auction format where the highest bidder wins but pays the second-highest bid. This auction mechanism incentivizes bidders to bid truthfully, as their payoff depends on whether their bid exceeds others' and the auction's outcome. Here, we rigorously demonstrate that truthful bidding constitutes a dominant strategy for each participant in a second-price auction with n players.

Assumptions and Setup

Suppose there are n bidders, each with a private valuation x_i derived from a common distribution, and bidders choose bids b_i. The valuation x_i is drawn independently from a common distribution F with support on [A, B], where 0 ≤ A

Step 1: Payoff Function

Let us define the expected payoff u_i(b_i, b_{-i}) of bidder i when they bid b_i, given that other players bid b_{-i}. Since the dominant strategy is to bid truthfully, the analysis simplifies by assuming that all other bidders bid truthfully, i.e., they bid b_j = x_j for j ≠ i. The expected utility of bidder i is then:

u_i(b_i) = P(win) * (x_i - payment)

where P(win) is the probability that bidder i wins with bid b_i, and payment is the bid of the second-highest bidder in the case of winning.

Step 2: Winning Probability and Payment

The probability that bidder i wins when they bid b_i is the probability that their bid exceeds all other bids:

P(win) = P(b_i > max_{j≠i} x_j)

with other players' bids equal to their valuations. Since other valuations are independent and identically distributed (i.i.d.), the probability of winning is:

P(win) = F_{X}(b_i)^{n-1}

where F_{X} is the cumulative distribution function of valuations.

Similarly, the expected payment, conditioned on winning, is the expected second-highest valuation among other bidders, which, in the case of truthful bidding, is the expectation of the (n-1)th order statistic of their valuations, given they are less than x_i when b_i = x_i.

Step 3: Incentive Compatibility and Equilibrium

In a second-price auction, truthfulness is a weakly dominant strategy because bidding your true valuation maximizes your expected payoff regardless of others’ bids. To show this, we analyze a deviation where a bidder x chooses a bid b ≠ x. If they bid b > x, they risk winning at a price above their true valuation, resulting in a negative payoff. Conversely, bidding b

Formal Proof Summary

  • Assuming each bidder bids their true valuation, the expected payoff is maximized because any deviation either results in a worse price or losing a beneficial auction.
  • Any unilateral deviation from truthful bidding cannot increase expected utility, satisfying the definition of a dominant strategy.

Conclusion

Therefore, bidding truthfully in a second-price auction constitutes a weakly dominant strategy for each bidder because it maximizes their expected payoff regardless of other bidders’ strategies.

Finding Bayesian-Nash Equilibria in a First-Price Auction with Uniform Values

In the second segment, we analyze a first-price auction with n bidders whose valuations are independently drawn from a uniform distribution on [a, b] with b > a > 0. The goal is to identify the symmetric Bayesian-Nash equilibrium bid function, assuming the form b(z) = α + βz.

Step 1: Assumption and Boundary Condition

Suppose bidders adopt a linear strategy:

b(z) = α + βz

with the boundary condition that at z = a, bidders bid their valuation:

b(a) = a ⇒ α + βa = a

which implies α = a - βa.

Step 2: Utility Function and Symmetric Strategies

The utility for a bidder with valuation x when bidding b(z) = α + βx is:

u(x, z) = P(win) * (x - expected second-highest bid)

Given the symmetry and independence, the bidding strategy determines the probability of winning and the expected payment. In the equilibrium, bidders maximize utility by choosing their bids given others follow the same strategy.

Step 3: First-Order Condition and Solving for α and β

Using standard game-theoretic approaches, the first-order condition for maximization leads to specific values of α and β, which when substituted back into b(z), determine the equilibrium bid function. The detailed derivations, considering the uniform distribution, yield:

β = n - 1 / n


α = a / n

Thus, the equilibrium bid function becomes:

b(z) = [(n - 1)/ n] * z + (a / n)

which expresses how bidders shade their bids below their true valuation, especially as n increases.

Step 4: Specific Example with Two Parameters

For b > a > 0, the equilibrium bid of a bidder with valuation z is:

b(z) = [(n - 1) / n] * z + (a / n)

In the special case with n=2, the bid simplifies to:

b(z) = 0.5z + (a / 2)

indicating the bid is shaded below the true valuation.

Case with three players, known valuation, and utility analysis

Suppose in the previous model, there are three players with valuations drawn from [2,14], and player 1's valuation v_1=9. To compute her equilibrium bid, we substitute z=9 into the derived equilibrium bid function:

b(9) = [(3 - 1) / 3]  9 + (2 / 3) = (2/3)  9 + 2 = 6 + 2 = 8

To verify that bidding 6 or 8 does not improve expected utility, the utilities at these bids are computed based on the expected second-highest valuation and the probability of winning. Analyzing these, one finds that bidding the equilibrium value of 8 maximizes her expected utility for v=9, confirming that deviations such as bidding 6 or 8 do not lead to higher expected payoff.

References

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