B. Expert-Defined Values

Loki uses several hard-coded constants where the optimal value is too difficult to determine (or we have not invested the effort to determine a better value). They have been selected by a poker expert, Darse Billings, and have not been experimentally validated. They are used as placeholders, and we foresee eventually upgrading the system so that the computer can determine proper behavior on its own.

For the post-flop betting strategy, there are only two expert-defined values: make1 and make2 (these are absolute constants, 0.50 and 0.85 respectively, not to be confused with Make1 and Make2). These are the betting and raising thresholds for EHS', and represent the likely hand strength required for these actions to be profitable.

For the pre-flop, when the function SetThresholds is called, it is given three parameters: group, tightness and position. The first two parameters are used to retrieve a set of values ([ base, increment], one per strategy). Then the function generates a threshold for each strategy using Equation 6.1, which in turn uses position.

• probability_play: This value is hard-coded at 0.6, meaning when estimating the number of players (for group) we expect 60% of all players, who have not yet acted, to play to the flop.

• group: For the pre-flop, we have reduced the number of classifications, based on the number of players, to three groups: 2, 3-4 and 5 or more players. We consider these the most important groupings and the IR rankings for more than 4 players are nearly identical in any case.

• tightness: There are three settings ( tight, moderate and loose). By default Loki plays with the loose setting. Note that all three sets of parameters are relatively tight styles of play. Even the loose set of parameters is considerably more conservative than a typical loose human player.

• strategy: There are several different pre-flop strategies that we may use: Make0, Call1, Make1, Call2, Make2 and Make4. Each strategy has a different threshold value based on the context, except Make0 which is the folding strategy. The values for Call1 and Make1 are the same, as are the values for Call2 and Make2, except for the case of the small blind, which has fixed values for these two strategies depending only on group.

• base, increment: There is one pair of [ base, increment] values defined per group, strategy and tightness (presented in Table B.1). For the small blind, the Call1 threshold is fixed at 0 for 5 or more players, -75 for 3-4 players, and equal to the Make1 threshold for 2 players. The Call2 threshold is 450 for 5 or more players, 200 for 3-4 players, and equal to the Make2 threshold for 2 players.

  
  

  Table B.1: Values for [ base, increment]

  2 players
  	tight	moderate	loose
  Make1	(-50,50)	(-50,50)	(-50,50)
  Make2	(150,50)	(50,50)	(0,0)
  Make4	(300,0)	(300,0)	(300,0)
  3-4 players
  	tight	moderate	loose
  Make1	(50,50)	(50,25)	(50,10)
  Make2	(200,50)	(200,25)	(200,10)
  Make4	(580,0)	(580,0)	(580,0)
  5 or more players
  	tight	moderate	loose
  Make1	(0,70)	(0,50)	(0,30)
  Make2	(450,50)	(450,25)	(450,10)
  Make4	(900,0)	(900,0)	(900,0)

  
  

For opponent modeling, we use expert values in the re-weighting function.

• Default Frequencies: In the absence of sufficient data, the action frequencies are weighted by hard-coded default frequencies (Equation 7.4). The set of values for d'[x][a] are presented in Table B.2.
• Re-weighting Function $\sigma$: As described in Section 7.2, for the pre-flop we use a $\sigma$ of 330. For the post-flop we use $\sigma = 0.4 * (1 - \mu)$, to reflect the fact that tighter players adhere to a narrow range of threshold hands.

Table B.2: Default frequencies d'[x][a]

	action a
bets to call x	Fold	Check/Call	Bet/Raise
0	0	0.5	0.5
1	0.5	0.3	0.2
2+	0.7	0.2	0.1