6.1 Pre-Flop Betting Strategy

The implemented pre-flop betting strategy is preliminary and makes use of expert information. It is sophisticated enough to not hamper overall performance so that the focus can be put on post-flop play (the more interesting portion of the game). Loki examines several variables, uses the expert knowledge to determine thresholds for various betting actions, and chooses a strategy:

• Make0: fold if it costs more than zero to continue playing, otherwise check.
• Call1: fold if it costs two or more bets to continue (and we have not already voluntarily put money in the pot this round), otherwise check/call.
• Make1: like Call1 except bet if there has not been a bet this round (with the big blind this cannot happen in the pre-flop).
• Call2: always call/check (despite what the name of this strategy suggests, even more than 2 bets).
• Make2: bet/raise if less than two bet/raises have been made this round, otherwise call.
• Make4: bet/raise until the betting is capped, otherwise call.

Except for the small blind, which is given special treatment, Call1 and Call2 are effectively not used, so there are really only 4 different strategies. Once a strategy is selected it is fixed for all subsequent actions in the pre-flop. The small blind is a special case due to only having to call one half of a bet (so Call1 is really Call0.5 and Call2 is really Call1.5), and has fixed thresholds for these two strategies.

Call1 (and hence Make1) has a special case folding requirement, that ``we have not already voluntarily put money in the pot this round." This is a feature added after testing with human opponents on IRC. Many players were very aggressive and would raise frequently. This meant that often when Loki called the blind with a decent hand (but not Call2 or better), two or more players would then raise, causing Loki to fold. Due to this commonly exploited weakness, the kludge is necessary until some amount of opponent of modeling is implemented into the pre-flop.

The thresholds for selecting a betting strategy are determined from a set of linear formulas of the form

threshold(strategy) = base + increment * position (6.1)

where strategy is the betting strategy ( e.g. Make1), base and increment are defined by a human expert (see Appendix B), and position is the number of players to act before it is the small blind's turn again (so, if $num\_inpot$ is the number of players still in the pot, the small blind is position $num\_inpot - 1$, the big blind is position $num\_inpot - 2$, the first player to act is position $num\_inpot - 3$ and so on, until the button (dealer) who is position 0).

For example, if Loki is playing tight against 3-4 players, the [ base, increment] values used for Make2 are [200,50] (based on Table B.1) so the formula used is

$$threshold({\bf Make2}) = 200 + 50*position$$

based on Equation 6.1. That is, the coefficients used for the linear formula depend on both the variable (group) and the parameter (tightness).

The variable group is based on the expected number of players ( $E\_num\_players$), which is

$$E\_num\_players num\_guaranteed +$$ = (6.2)

$$probability\_play * (num\_inpot - num\_guaranteed).$$

There are three cases of interest for the expected number of players. We round $E\_num\_players$ to the nearest integer and determine the appropriate group based on what range that value falls in: ``2 players", ``3-4 players" and ``5 or more players" (see Appendix B). The state variable $num\_guaranteed$ is the number of players who have already put money in the pot (and have presumably committed to playing). This includes the blinds as well as ourselves (we assume we will play). $Probability\_play$ is an expert-defined value for the average playing percentage of players (by default 0.60). Appropriate opponent modeling would provide much better estimates for this value, based on the observation of the current session .

The parameter tightness is a setting which affects the percentage of hands that Loki will play (indirectly, by selecting a different set of thresholds). The three settings are tight, moderate, and loose (the default). With ten players these roughly translate into playing 18%, 21% and 24% of all hands (the distinction is not large so the terms are a misnomer - all levels are relatively conservative).

There is one [base, increment] pair per set of group, strategy and tightness values (Table B.1) so there are 27 pairs total (since Make0 is the default strategy only three thresholds are needed for determining the strategy). Once the thresholds are determined, the actual strategy selected is dependent on the pre-calculated income rate (IR) of the hole cards. Figure 6.1 describes the algorithm for selecting a strategy.

For example, consider a six player game. We hold A$\diamondsuit$-T$\diamondsuit$ and are playing tight. The first two players put in blinds, the next two fold, and it is Loki's turn. There is still one player, the button, who has not yet acted. So position = 1, $num\_inpot = 4$ and $num\_guaranteed = 3$ (including Loki). We then calculate

$$E\_num\_players = 3 + 0.60 * (4 - 3) = 3.6$$

and use this to determine group = ``3-4 players". Then, using Table B.1 we calculate

$$\begin{eqnarray*} threshold({\bf Make1}) =& 50 + 50*1 &= 100, \\ threshold({\bf... ...tnormal{and} \\ threshold({\bf Make4}) =& 580 + 0*1 &= 580. \\ \end{eqnarray*}$$

Finally, we find the value of our hand is IR = 491 and select the appropriate strategy. Since $250 \le 491 < 580$, we select the Make2 strategy, and raise.

Figure 6.1: Pre-Flop Betting Strategy