A Silverman game is a two-person zero-sum game defined in terms of
two sets S I and S II of positive numbers, and two parameters, the
threshold T > 1 and the penalty v > 0. Players I and II
independently choose numbers from S I and S II, respectively. The
higher number wins 1, unless it is at least T times as large as the
other, in which case it loses v. Equal numbers tie. Such a game
might be used to model various bidding or spending situations in
which within some bounds the higher bidder or bigger spender wins,
but loses if it is overdone. Such situations may include spending
on armaments, advertising spending or sealed bids in an auction.
Previous work has dealt mainly with special cases. In this work
recent progress for arbitrary discrete sets S I and S II is
presented. Under quite general conditions, these games reduce to
finite matrix games. A large class of games are completely
determined by the diagonal of the matrix, and it is shown how the
great majority of these appear to have unique optimal strategies.
The work is accessible to all who are familiar with basic
noncooperative game theory.
General
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