Constructive comparison in bidding combinatorial games

Discrete Richman Bidding Combinatorial Games that generalize alternating normal play were introduced by Kant, Larsson, Rai, and Upasany (2024). The major questions concerning defined outcomes were resolved. By generalizing standard techniques from alternating play, Conway (ONAG, 1976), Berlekamp et al. (Winning Ways, 1982) and Siegel (2013), we study an algorithmic play-solution to the problem of game comparison. We demonstrate some consequences of this result that generalize classical alternating play results. In particular, integers, dyadics and defined zugzwangs have many nice properties, such as group structures, but on the other hand a nim heap of size one \(*=\left\{ 0\!\mid \!0\right\} \) $\ast =\left\{0,,\mid ,,0\right\}$ becomes non-invertible in case of non-trivial bidding. Moreover, we state a couple of thrilling conjectures and open problems for readers to dive into this promising path of bidding combinatorial games.