Two auction mechanisms are studied in which players compete with one another for an exogenously determined prize by independently submitting integer bids in some discrete and commonly known strategy space specified by the auctioneer. In the unique lowest (highest) bid auction game, the winner of the prize is the player who submits the lowest (highest) bid provided that this bid is unique, i.e., unmatched by other bids. Assuming a commonly known finite population of players, a non-negative cost of entry, and an option to stay out of the auction if the entry cost is deemed too high, we propose an algorithm for computing symmetric mixed-strategy equilibrium solutions to the two variants of the auction game, illustrate them, and examine their properties.
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