# Discrete All-Pay Bidding Games

@article{Menz2015DiscreteAB, title={Discrete All-Pay Bidding Games}, author={Michael Menz and Justin Wang and J. Xie}, journal={ArXiv}, year={2015}, volume={abs/1504.02799} }

In an all-pay auction, only one bidder wins but all bidders must pay the auctioneer. All-pay bidding games arise from attaching a similar bidding structure to traditional combinatorial games to determine which player moves next. In contrast to the established theory of single-pay bidding games, optimal play involves choosing bids from some probability distribution that will guarantee a minimum probability of winning. In this manner, all-pay bidding games wed the underlying concepts of economic… Expand

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+ 1) · λ(0, s 0 , . . . , s −1 ) + (1 − λ)(s 0 , . . . , s −1 , 0) = v A 1

- + 1) · λ(0, s 0 , . . . , s −1 ) + (1 − λ)(s 0 , . . . , s −1 , 0) = v A 1

B's payoff x against against A bidding + 1 will be x = 1 − (α +1 s + α 1 s 0 )

- B's payoff x against against A bidding + 1 will be x = 1 − (α +1 s + α 1 s 0 )

By our assumption, v > v A so S is strictly better than S A against R(S A ). Thus S A cannot be a Nash Equilibrium strategy, which is a contradiction

- By our assumption, v > v A so S is strictly better than S A against R(S A ). Thus S A cannot be a Nash Equilibrium strategy, which is a contradiction

Note that because player A is winning ties, α +1 < α , . . . , α 1 < α 0 , as in each case A is winning by one more chip. Thus, α s + · · · + α 0 s 0 > α +1 s + α 1 s 0 which means

- Note that because player A is winning ties, α +1 < α , . . . , α 1 < α 0 , as in each case A is winning by one more chip. Thus, α s + · · · + α 0 s 0 > α +1 s + α 1 s 0 which means

Since B bids at most − 1, we only need to consider the first coordinates of

- Since B bids at most − 1, we only need to consider the first coordinates of