Revenues in discrete multi-unit, common value auctions: a study of three sealed-bid mechanisms

Abstract

This paper proposes a discrete bidding model for both quantities and pricing. It has a two-unit demand environment where subjects bid for contracts with an unknown redemption value, common to all bidders. Prior to bidding, the bidders receive private signals of information on the (common) value. The relevant task is to compare the equilibrium strategies and the seller’s revenue of the three most common auction formats with two players. The result is that the Vickrey auction always gives the most revenue to the seller, the discriminatory auction follows closely and the uniform auction clearly is the worst due to demand reduction.

Appendix

Proof

The equilibrium prescribes equal bids on both units and the payoff is:

$$ \begin{array}{rcl} \pi\big(b^*\big) &=& \frac{1}{12} \sum_{j=1}^{t_i - 1} 2 \left[(t_i+j\,) - \left(t_i + \left\lceil \frac{t_i}{3} \right\rceil \right) \right] + \frac{7}{12} \left[ \left(\frac{3}{2}t_i + \frac{7}{4}\right) - \left(t_i + \left\lceil \frac{t_i}{3} \right\rceil \right) \right] \\ &=& \frac{1}{12} \sum_{j=1}^{t_i - 1} 2 \left[j - \left\lceil \frac{t_i}{3} \right\rceil \right] + \frac{1}{12} \left[\frac{49}{4} + \frac{7}{2}t_i - 7 \left\lceil \frac{t_i}{3} \right\rceil \right] \\ &=& \frac{1}{12} \sum_{j=1}^{t_i - 2} 2 \left[j - \left\lceil \frac{t_i}{3} \right\rceil \right] + \frac{1}{12} \left[\frac{41}{4} + \frac{11}{2}t_i - 9 \left\lceil \frac{t_i}{3} \right\rceil \right] \end{array} $$

Now, suppose she deviates and bids b′(t _i ) = b(t _i  − l), where l ∈ ℤ₊, i.e, she mimics a type-(t _i  − l) type. Then, she will always lose both units when playing against another type-t _i bidder and, depending on which type she mimics, both will win one unit each when playing against each other. It shows that the best she can do is to mimic a type-(t _i  − 1) player, but it is not good enough since she will then only get payoff:

$$ \begin{array}{rcl} \pi\big(b'\big) &=& \frac{1}{12} \sum_{j=1}^{t_i - 2} 2 \left[(t_i+j\,) - \left(t_i - 1 + \left\lceil \frac{t_i - 1}{3} \right\rceil \right) \right] \\ &&+ \frac{1}{12} \left[ (t_i + t_i - 1) - \left(t_i - 1 + \left\lceil \frac{t_i - 1}{3} \right\rceil \right) \right] \\ &=& \frac{1}{6} \sum_{j=1}^{t_i - 2} \left[j + 1 - \left\lceil \frac{t_i - 1}{3} \right\rceil \right] + \frac{1}{12} \left[ t_i - \left\lceil \frac{t_i - 1}{3} \right\rceil \right] \end{array} $$

Which shows, if we use the relation: $\lceil \frac{t_i}{3} \rceil - 1 \leq \lceil \frac{t_i - 1}{3} \rceil$, that:

$$ \begin{array}{rcl} \pi\big(b^*\big) - \pi\big(b'\big) \geq \frac{1}{12} \left(\frac{69}{4} + \frac{1}{2} t_i - 8 \left\lceil \frac{t_i}{3} \right\rceil \right) > 0, \qquad \forall t_i \in T_i. \end{array} $$

If, on the other hand, the player deviates by raising the bid compare to the equilibrium bid, and bids b′(t _i ) = b(t _i  + l), where l ∈ ℤ₊, i.e, she mimics a type-(t _i  + l) type. Then, she will always win both units when playing against another type-t _i bidder and, depending on which type she mimics, both will win one unit each when playing against each other. It shows that the best she can do is to mimic a type-(t _i  + 1) player, but it is not good enough since she will then only get payoff:

$$ \begin{array}{rcl} \pi\big(b'\big) &=& \frac{1}{12} \sum_{j=1}^{t_i - 1} 2 \left[(t_i+j\,) - \left(t_i + 1 + \left\lceil \frac{t_i + 1}{3} \right\rceil \right) \right] \\ &&+\frac{7}{12} 2 \left[ \left(\frac{3}{2}t_i + \frac{7}{4}\right) - \left(t_i + 1 + \left\lceil \frac{t_i + 1}{3} \right\rceil \right) \right] \\ &&+ \frac{1}{12} \left[ (t_i + t_i + 1) - \left(t_i + 1 + \left\lceil \frac{t_i + 1}{3} \right\rceil \right) \right] \\ &=& \frac{1}{12} \sum_{j=1}^{t_i - 1} 2 \left[j - 1 - \left\lceil \frac{t_i + 1}{3} \right\rceil \right] + \frac{14}{12} \left[\frac{1}{2} t_i + \frac{3}{4} - \left\lceil \frac{t_i + 1}{3} \right\rceil \right] \\ &&+ \frac{1}{12} \left[t_i - \left\lceil \frac{t_i + 1}{3} \right\rceil \right] \end{array} $$

If we use the relation: $\lceil \frac{t_i}{3} \rceil \leq \lceil \frac{t_i + 1}{3} \rceil$, we have:

$$ \begin{array}{rcl} \pi\big(b^*\big) - \pi\big(b'\big) \geq \frac{1}{12} \left(- \frac{1}{4} - \frac{5}{2} t_i + 8 \left\lceil \frac{t_i}{3} \right\rceil \right) > 0, \qquad \forall t_i \in T_i. \end{array} $$

Thus, no one has any incentives to deviate from the proposed equilibrium strategy. □