Gerald A. Heuer

Silverman's game on discrete sets

Abstract In a symmetric Silverman game each of the two players chooses a number from a set S ⊂(0,∞). The player with the larger number wins 1, unless the larger is at least T times as large as the other, in which case he loses v . Such games are investigated for discrete S , for T >1 and v >0. Except for v too near zero, where there is a proliferation of cases, explicit solutions are obtained. These are of finite type and, except at certain boundary cases, unique.

Three-part partition games on rectangles

A partition game on a rectangle is a two-person zero-sum game in which the rectangle is partitioned into a finite number of regions and the payoff function is constant in each region. A special case with four regions is essentially Silvermans game, which has been extensively studied in recent years. In this paper, we examine three-part partition games where the boundaries separating the regions are the graphs of two continuous functions. Various conditions under which optimal mixed strategies and game value do or do not exist are found, as are optimal strategies and value in case they exist. Some games of this type have in the classical game theory literature been interpreted as continuous Colonel Blotto games.

Which four-part partition games are isomorphic to Silverman's game?

Abstract A four-part partition game on a rectangle is a two-person zero-sum game where the pure strategy sets are intervals and the resulting rectangle is partitioned by three curves into four regions, on each of which the payoff is constant. Such a game is essentially a generalization of Silvermans game, which has been studied extensively by the author and others. It has been shown previously that if two of the partitioning curves of suitable type are given, and the four payoffs are suitably related, there is a unique choice of the third curve for which the game is isomorphic to Silvermans game. In the present note this third curve is expressed explicitly in terms of the first two, provided that one of the two given curves is the middle one of the three.

Four-part partition games isomorphic to Silverman's game

Abstract A four-part partition game on a rectangle S I × S II is a two-person zero-sum game, where the strategy sets S I and S II are intervals and the rectangle is partitioned by three curves into four regions on each of which the payoff function is constant. These games generalize Silvermans game, where the boundary curves are of the form y = Tx , y = x and y = x / T . In this paper it is shown that two increasing nonintersecting curves may be chosen arbitrarily and there is a uniquely determined third increasing curve such that if the payoffs are properly related, the game is isomorphic to Silvermans game.

Balanced 3 by 3 games

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 W1 and W2 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.

Silverman’s Game on Intervals: Preliminaries

The symmetric Silverman game on intervals was analyzed by R. J. Evans in the 1979 article [1]. We shall obtain Evans’ results as a special case of our results for general intervals in Chapters 2–4, especially in the Section on Case 1 in Chapter 3 (pp.47–49), but it is interesting and useful to look at Evans’ results first, and later to see how they are part of a larger picture. When S I = S II the game is completely symmetric, so that if a game value V exists it must be the case that V = 0, and every optimal strategy for one player is optimal for the other as well.

Intervals with Equal Left Endpoints or Equal Right Endpoints

The first group of theorems of this chapter will allow us to show that no game value exists in the regions H n of Figure 1 if S I and S II have equal left endpoints and none exist in regions LA n if they have equal right endpoints. In particular, from Theorems 3.1, 3.4 and 3.10 it follows that there are no optimal strategies in the symmetric game on open intervals when (2.0.1) fails. Under conditions where optional strategies do exist, we shall usually find them among the key mixed strategies introduced in Chapter 2.

The Disjoint Discrete Game

The disjoint game was rather thoroughly investigated in [13], where it was shown that when v ≥ 1, the discrete games fall into eight classes called 1A, 2A, 2B, 3A, 3B, 4A.k, 4B.k and 5A.k. Class 2B becomes 2A if the strategy sets are interchanged, and the same is true of the pairs {3A, 3B} and {4A.k, 4B.k}. It was assumed in [13] that c1 ≤ d1 in the original strategy sets S I and S II , but sometimes in the reduced game the first element of Player I’s strategy set would be larger than the first element in the other. Less complete results were obtained for v < 1. Some of these will be presented later in this chapter.

The Symmetric Discrete Game

In this chapter and the next we apply the results of the preceding chapters on Silverman’s game on discrete sets to the cases S I = S II (the symmetric game) and S I ∩ S II = O (the disjoint game), and give some further results in these special cases. In all of these cases we will obtain optimal strategies and game values explicitly. In some of them we give some additional results for v < 1.

The Further Reduction of Semi-Reduced Games

In the sequel we shall frequently consider a block, or run, of successive elements \(E = \{ {{e}_{k}},{{e}_{{k + 1}}}, \ldots ,{{e}_{q}}\}\) from \({{\widetilde{W}}_{I}}\) versus a block \(F = \{ {{f}_{l}},{{f}_{{l + 1}}}, \ldots ,{{f}_{r}}\}\) from \({{\widetilde{W}}_{{II}}}\), and argue that against the elements of F a certain emin E dominates. The meaning is that in the subgame on E × E, every ei is dominated by em; i.e., that