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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.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract
A Silverman game is a two person zero sum game defined in terms of two sets, S1 and S2, of positive numbers and two parameters, the threshold T > 1 and the penalty ν > 0. Players I and II choose numbers independently from S1 and S2, respectively. The higher number wins 1, unless it is at least T times as large as the other, in which case it loses ν. If the numbers are equal the payoff is zero.
Gerald A. Heuer, Ulrike Leopold-Wildburger

2. Games with saddle points

Abstract
The theorems in [7] dealing with classes 1A, 2A and 2B do not depend on the strategy sets being disjoint, and include all Silverman games where at least one player has an optimal pure strategy, except the symmetric 1 by 1 case:
  • THEOREM 2.1. In the symmetric Silverman game (S,T,ν), suppose that there is an element c in S such that c < Tci for all ci in S, and that S n (c,Tc) = Φ. Then pure strategy c is optimal.
  • PROOF. Let A(x,y) be the payoff function. By symmetry the game value is 0. Since A(c,y) = 1,0 or ν according as y < c, y = c or y ≥ Tc, we have A(c, y) ≥ 0 for every y in S.
Gerald A. Heuer, Ulrike Leopold-Wildburger

3. The 2 by 2 games

Abstract
For the remainder of the paper we assume that S1 and S2 are discrete. It turns out that a great many discrete Silverman games are reducible to 2 by 2 games, in the sense that each player has a 2-component optimal mixed strategy. In this section we shall identify all irreducible 2 by 2 Silverman games, and in the next section are some theorems giving conditions under which games reduce to 2 by 2. “Game” hereafter will always mean “Silverman game.”
Gerald A. Heuer, Ulrike Leopold-Wildburger

4. Some games which reduce to 2 by 2 when v ≥ 1

Abstract
The game of case (A) above and its dual (A’) are the reduced games of Classes 3A and 3B in the disjoint case [7]. However, many games where S1 n S2Φ also reduce to these 2 by 2 games, as we see in the first two theorems below. From now on we assume also that ν ≥ 1.
Gerald A. Heuer, Ulrike Leopold-Wildburger

5. Reduction by dominance

Abstract
In [6], it is shown that every discrete Silverman game with ν ≥ 1 reduces by dominance to a finite game, and in [7], it is shown that if Si ∩ [a,b] = Φ, where a and b are elements of S3-i, then b is dominated by a. In this section we shall discuss four types of dominance for Silverman games, including the above two. Through repeated reduction of the strategy sets S1 and S2 by means of these four types of dominance we obtain what we call pre-essential sets W̃1 ⊂ S1 and W̃2 ⊂ S2. These are minimal subsets in the sense that no further reduction is possible through the use of these four types of dominance.
Gerald A. Heuer, Ulrike Leopold-Wildburger

6. Balanced 3 by 3 games

Abstract
When n = 1 the pre-essential sets have three elements each. There are nine different possible diagonals, and none of these games reduces further. Thus W͂1 and W͂2 are already the essential sets. The nine diagonals and the solutions of the corresponding 3 by 3 games are given below. We abbreviate the diagonal elements -1 and +1 by - and +, respectively. P = (p1, p2,p3) is the optimal strategy for Player I, Q = (q1,q2,q3) that for Player II. V is the game value.
Gerald A. Heuer, Ulrike Leopold-Wildburger

7. Balanced 5 by 5 games

Abstract
Subject to our restriction that the first nonzero diagonal element is -, there are exactly 50 balanced 5 by 5 games. We may list them in lexicographic order of diagonals from 0 0 0 0 0 to - - - - - (with the ordering 0 < - < +). Of these fifty, the five with diagonals of the form — 0 + x y reduce to 2 by 2 games of type A, as may be seen from Theorem 10.1 below. They are numbers 34–38 in our ordering. The four with diagonals x y — 0 + similarly reduce to 2 by 2 games of type A’, as implied by Theorem 10.2. They are numbers 7, 19, 31 and 48. The four having diagonals — x 0 y +, numbers 24, 28, 41 and 45, reduce to 3 by 3, as implied by Theorem 8.1.
Gerald A. Heuer, Ulrike Leopold-Wildburger

8. Reduction of balanced games to odd order

Abstract
Recall that for balanced Silverman games the payoff matrix is completely determined by the diagonal, and that every diagonal element is 1, 0 or -1. The evidence strongly suggests that unless both 1 and -1 occur (and therefore all three of 1, 0, -1), the game is irreducible. If both 1 and -1 occur, with one of them in the middle position, then the game reduces to 2 by 2, as we show in Section 10. In this section and the next three, we examine the reduction for all other diagonals; i.e. those where each of 1, 0 and -1 occur on the diagonal and the middle element is 0. Those which reduce to an odd order game are treated in the present section and those reducing to even order in Section 9.
Gerald A. Heuer, Ulrike Leopold-Wildburger

9. Reduction of balanced games to even order

Abstract
In this section we describe the reduction of the remaining eighteen of the 36 cases in (8.0.3), (8.0.4) and (8.0.7). There are again four types of reduced game, corresponding to (A), (B), (C) and (D) in (8.0.4). In our description of these, the first nonzero main-diagonal element is again always -1, and off-diagonal zeros are concentrated in a middle segment of the first subdiagonal. The remainder of th€ matrix is the same in all cases, and may be described by the diagram in Figure 9.
Gerald A. Heuer, Ulrike Leopold-Wildburger

10. Games with ±1 as central diagonal element

Abstract
When the central diagonal element is ±1, the facts are considerably simpler. It again appears to be the case that unless both +1 and -1 occur on the diagonal, the game is irreducible. We shall show that when both do occur, the game always reduces to the 2 by 2 game \( \left[ {\begin{array}{*{20}{c}} { - 1}&v \\ 1&{ - 1} \end{array}} \right]{\text{ }}or{\text{ }}\left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ { - v}&1 \end{array}} \right] \) according as the central diagonal element is +1 or -1. Let us denote the diagonal elements (x1, x2,..., x2n+1).
Gerald A. Heuer, Ulrike Leopold-Wildburger

11. Further reduction to 2 by 2 when v = 1

Abstract
We show now how all of the reduced games in Sections 8 and 9 reduce further, if v = 1, to 2 by 2 games with matrix
$$ {A_0} = \left[ {\begin{array}{*{20}{c}} 1&{ - 1} \\ { - 1}&1 \end{array}} \right]. $$
(11.0.1)
.
Gerald A. Heuer, Ulrike Leopold-Wildburger

12. Explicit solutions for certain classes

Abstract
In the papers [2] on symmetric games and [7] on disjoint games, explicit optimal strategies and game values are obtained for all games. The fact that the diagonal consists entirely of zeros in the symmetric case and entirely of ones in the disjoint case has the effect that the components in the optimal strategy vectors may be described by simple recursions. For nonconstant diagonals these relations among the components are less regular, but in a few cases where the diagonal is nearly constant one can still obtain relatively nice explicit formulas. We shall do so here for diagonals which are constant except for the middle element, or constant except for the last element.
Gerald A. Heuer, Ulrike Leopold-Wildburger

13. Concluding remarks on irreducibility

Abstract
We conclude with brief remarks about the evidence that the reduced games obtained in Sections 8 and 9 are not further reducible. (Those in Sections 10 and 11 clearly are not.)
Gerald A. Heuer, Ulrike Leopold-Wildburger

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