### Discussion Paper

No. 2017-22 |
May 08, 2017

Nash equilibria in all-pay auctions with discrete strategy space

## Abstract

Using two-player all-pay auctions, the author fully characterizes the Nash equilibrium under a discrete bidding strategy space. In particular, he shows that under the random tie-breaking rule, the cardinality of the set of Nash equilibrium depends on the parity of the reward size and a continuum of Nash equilibria exists. Additionally, when a simple favor-one-sided tie-breaking rule is used, the equilibrium solution becomes independent of the reward size.

## Comments and Questions

This is an interesting paper and the author derives relevant results. However, I believe the paper may benefit from some important changes in its exposition.

In general, the results are only described in mathematical terms and no intuition or description in words is provided (see e.g. the paragraph after Prop ...[more]

... 2). This does not facilitate the reader's understanding, and the author should exert more effort guiding the reader towards a good understanding of the results.

The author states that most theoretical models assume a continuous strategy space for tractability concerns. Despite a continuous strategy space often provides a more tractable setting, tractability does not have to be the main reason for this choice by the literature, and it is not clear what strategy space (continuous or discrete) is more suitable for each application.

When the author introduces the condition Q=2n, the variable n has not been defined. If n is just a natural number, the author should mention that the condition requires Q to be an even number.

P's and V's of Proposition 1 are not defined. This sharply breaks the flow of the paper, because it forces the reader to look for their definition. The original papers from which part of the result is taken (Bouckaert et al 1992 and Schep 1985) are not written in English, and thus they do not help understand the author's notation. The explanation of the Proposition at page 3 clarifies (too late) the meaning of the notation.

A graphical representation of the distribution of mass of Proposition 1 might greatly help the reader understand the results.

After Prop 1, a comparison with the case of continuous strategy space might be helpful.

At page 4, "When Q_y=2n+1, it is always [...]" lacks the conclusion that one should draw from that inequality.

The author repeatedly says that the Nash equilibrium depend on the parity of the reward size. The meaning of this remains cryptic until one enters into the details of the paper. I suggest clarifying from the beginning its meaning, rephrasing for instance to: the valuation of the prize is an odd or an even number.

In the proofs, the author uses "suggests" for formal results, such as "Lemma 4 suggests V=0". I believe that "implies" is a more appropriate word for analytical results. Additionally, “we get” should probably be substituted by “we obtain”.

All in all, the paper seems to provide interesting and novel results, but the poverty of the exposition probably does not give justice to the relevance of these results. I deem necessary a sharp exposition improvement before I will be able to express my final opinion on whether the paper is publishable.

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