Dice poker strategies

Poker s kockami je igra za dva igralca, ki jima je na začetku igre z metom kocke določena njuna vrednost, na podlagi katere se odločata kako igrati. Cilj naloge je najti strategijo, ki igralcu prinese največji dobiček. Začnemo s strategijami brez psiholoških iger, kjer se izkaže, da igralec $P_1$ v povprečju izgublja denar. Ko v strategijo igralca $P_1$ uvedemo blefiranje, njegov dobiček postane pozitiven in se še poveča, ko pri obeh igralcih dovoljujemo mešane strategije. Verjetnosti, s katerimi stavita igralca pri mešanih strategijah, dobimo s principom indiferentnosti, slabost takšne taktike pa je, da ugibamo, pri katerih vrednostih se igralca odločata med svojima akcijama. Nato preidemo na poker s kockami z naključnimi vrednostmi iz intervala $[0, 1]$ in spet preverjamo, kako igrata igralca ob določeni vrednosti. Izračunamo mejne vrednosti, kjer je igralcu vseeno katero akcijo uporabi, saj je njegovo izplačilo enako pri obeh akcijah in z njihovo pomočjo ugotovimo, kakšna je njuna strategija. Dobiček pri tej igri je še višji kot pri igri z mešanimi strategijami. Nato drugemu igralcu pustimo nadaljevati igro, če se prvi igralec na začetku igre odloči za odstop in poiščemo mejne vrednosti še tukaj. Izkaže se, da je v tem primeru število Nashevih ravnovesij odvisno od velikosti stav igralcev.Dice poker is a game for two players who are assigned their values at the beginning of the game by a throw of a dice and they decide how to play based on their values. The goal is to find a strategy that would bring the player the highest payout. We start with strategies without any psychological games, where it turns out that $P_1$ loses money on average. When we allow bluffing in his strategies his payout becomes positive and it increases even more, when we allow mixed strategies for both players. The probabilities with which the players bet in mixed strategies are obtained by principle of indifference. The weakness with such tactics is that we have to guess at which values the players have to decide between their actions. Then we continue with poker with random values from the interval $[0, 1]$ and again check how the players play with their assigned values. We find cutoffs, where the player doesn’t care which action he uses, because the payout is the same with both and with the help of those values we find out what their strategies are like. The payout in this game is even higher than in the mixed strategy game. Then the second player is allowed to continue the game if $P_1$ passes at the beginning of the game and we find the cutoffs again. It turns out that the number of Nash equilibrium in this case depends on the size of players’ bets