5.3.1 Multi-Player Considerations

As described in Section 5.2.1, Loki makes use of a field array provided by the opponent modeling module which is representative of the entire set of active opponents. Unlike hand strength, against multiple (two or more) opponents the PPOT and NPOT values calculated with the field array are used without adjustment. The correct calculation would be similar to the one described in Section 5.2.1 (accounting for each player adds an extra iteration layer of approximately 1,000 subcases), but, for potential, the value against one player is believed to be a simple but reasonable estimate (high usefulness and low computational complexity).

Calculating potential in the context of multiple opponents is complex due to the many interactions; there is no easy reduction to an approximate value similar to HS_n. However, it is likely that the present estimates are typically optimistic and the correct value would worsen with additional opponents. For example, consider the situation where you have a straight draw (4$\diamondsuit$-5$\clubsuit$/6$\heartsuit$-3$\diamondsuit$-T$\heartsuit$) but you are unaware of any tendencies of your opponents ( i.e. uniform weight arrays). Each additional opponent increases the chances that someone has a flush draw in hearts (in fact, it increases the chances of any particular hand occurring). This means that `intersection cards' like 2$\heartsuit$ and 7$\heartsuit$ are worth less to you since the likelihood that they give you the winning hand decreases, as does PPOT. For another example, consider that you hold top pair (K$\clubsuit$-T$\diamondsuit$/6$\heartsuit$-3$\diamondsuit$-T$\heartsuit$). Each additional opponent increases the chance that you are up against a flush or straight draw so cards like a 2, a 7 or a heart are more likely to give you a losing hand. Therefore, NPOT increases with additional opponents.

But over-optimism is not always the case; multiple opponents can improve your chances. For example, if you have a straight draw with small cards (4$\diamondsuit$-5$\clubsuit$/6$\clubsuit$-3$\diamondsuit$-K$\heartsuit$) and your opponents are likely holding high cards, then each additional opponent increases the chance that the upcoming card is not high. This will overall both decrease NPOT and increase PPOT, although additional players also increases the size of the pot, which increases the value of our draw.

These are only rough examples intended to demonstrate the complex interactions involved in multi-player considerations for hand potential. The actual effect of each additional opponent is dependent on the probability distribution of possible hands and its relationship to the probability distributions of the other players, as well as its relationship to your needed cards.