Guaranteed Scoring Games

The class of Guaranteed Scoring Games (GS) is the class of all two-player scoring combinatorial games with the property that Normal-play games (Conway et. al.) are order embedded into GS. They include, as subclasses, the scoring games considered by Milnor (1953), Ettinger (1996) and Johnson (2014). We present the structure of GS and the techniques needed to analyze a sum of guaranteed games. Firstly, GS forms a partially ordered monoid, via defined outcomes over the reals, and with disjunctive sum as the operation. In fact, the structure is a quotient monoid with partially ordered congruence classes. We show that there are four reductions that when applied, in any order, give a unique representative for each congruence class. The monoid is not a group, but in this paper we prove that if a game has an inverse it is obtained by ‘switching the players’. The order relation between two games is defined by comparing their outcomes in any disjunctive sum. Here, we demonstrate how to compare the games via a finite algorithm instead, extending ideas of Ettinger, and also Siegel (2013). ∗Supported by the Killam Trust. †Partially supported by NSERC ‡Work supported by centre grant (to BioISI, Centre Reference: UID/MULTI/04046/2013), from FCT/MCTES/PIDDAC, Portugal. the electronic journal of combinatorics 23(3) (2016), #P3.27 1