Participants enter the ride at their own risk. gord's sports centre racing ltd and affiliated sponsors will not be held responsible for any accidents or damage caused to any machines or individuals during the ride. all participants are responsible for the safety of their machines and their passengers. all drivers must have a valid motorcycle license and insurance coverage. by signing this waiver, you agree to the above and acknowledge you are a licensed and insured rider, and will abide by the rules of the road or 2005 world series of poker. We now describe how we use counterfactual regret minimization to compute a near equilibrium solution in the domain of poker. the 2005 world series of poker variant we focus on is heads-up limit texas hold'em, as it is used in the aaai computer poker competition [9]. the game consists of two players zero-sum ; , four rounds of cards being dealt, and four rounds of betting, and has just under 1018 game states [2]. as with all previous work on this domain, we will first abstract the game and find an equilibrium of the abstracted game. in the terminology of extensive games, we will merge information sets; in the terminology of poker, we will bucket card sequences. the quality of the resulting near equilibrium solution depends on the coarseness of the abstraction. in general, the less abstraction used, the higher the quality of the resulting strategy. hence, the ability to solve a larger game means less abstraction is required, translating into a stronger poker playing program. 4.1 abstraction the goal of abstraction is to reduce the number of information sets for each player to a tractable size such that the abstract game can be solved. early poker abstractions [2, 4] involved limiting the possible sequences of bets, e.g., only allowing three bets per round, or replacing all first-round decisions with a fixed policy. more recently, abstractions involving full four round games with the full four bets per round have proven to be a significant improvement [7, 6]. we also will keep the full game's betting structure and focus abstraction on the dealt cards. our abstraction groups together observed card sequences based on a metric called hand strength squared. hand strength is the expected probability of winning1 given only the cards a player has seen. this was used a great deal in previous work on abstraction [2, 4]. hand strength squared is the expected square of the hand strength after the last card is revealed, given only the cards a player has seen. intuitively, hand strength squared is similar to hand strength but gives a bonus to card sequences whose eventual hand strength has higher variance. higher variance is preferred as it means the player eventually will be more certain about their ultimate chances of winning prior to a showdown. more importantly, we will show in section 5 that this metric for abstraction results in stronger poker strategies. the final abstraction is generated by partitioning card sequences based on the hand strength squared metric. first, all round-one card sequences i.e., all private card holdings ; are partitioned into ten equally sized buckets based upon the metric. then, all round-two card sequences that shared a round-one bucket are partitioned into ten equally sized buckets based on the metric now applied at round two. thus, a partition of card sequences in round two is a pair of numbers: its bucket in the previous round and its bucket in the current round given its bucket in the previous round. this is repeated after reach round, continuing to partition card sequences that agreed on the previous rounds' buckets into ten equally sized buckets based on the metric applied in that round. thus, card sequences are partitioned into bucket sequences: a bucket from for each round. the resulting abstract game has approximately 1.65 1012 game states, and 5.73 107 information sets. in the full game of poker, there are approximately 9.17 1017 game states and 3.19 1014 information sets. so although this represents a significant abstraction on the original game it is two orders of magnitude larger than previously solved abstractions. 4.2 minimizing counterfactual regret now that we have specified an abstraction, we can use counterfactual regret minimization to compute an approximate equilibrium for this game. the basic procedure involves having two players repeatedly play the game using the counterfactual regret minimizing strategy from equation 8. after t repetitions of the game, or simply iterations, we return 1 , 2 ; as the resulting approximate t t t equilibrium. repeated play requires storing ri i, a ; for every information set i and action a, and updating it after each iteration.2 and las vegas poker rooms.
Or just rolldie 1: 6, true ; . many of the other sample space functions tosscoin ; , cards ; , roulette ; , etc. ; have similar makespace arguments. check the documentation for details. one sample space function that does not have a makespace option is the urnsamples ; function. this was intentional. the reason is that under the varied sampling assumptions the outcomes in the respective sample spaces are not, in general, equally likely. it is important for the user to carefully consider the experiment to decide whether or not the outcomes are equally likely, and then use probspace ; to assign the model. an unbalanced coin while the makespace argument to tosscoin ; is useful to represent the tossing of a fair coin, it is not always appropriate. for example, suppose our coin is not perfectly balanced, for instance, maybe the "h" side is somewhat heavier such that the chances of a h appearing in a single toss is 0.70 instead of 0.5. we may set up the probability space with probspace tosscoin 1 ; , probs c 0.7, 0.3 1 2 toss1 probs h 0.7 t 0.3. This chapter has outlined a proposed framework for assessing the security of a software process by testing it for the presence of vulnerabilities. the framework is composed of three components: 1 ; the object model, 2 ; the taxonomy of vulnerabilities, and 3 ; the v&v strategies component. the framework simplifies the testing of a software process for presence of vulnerabilities. the approach involves determining if the software process allows the violation of a constraint or an assumption; a vulnerability exists if either can be violated. furthermore, the framework provides a comprehensive solution for assessing security, because it can be applied to assess security of any software process executing above the level of the operating system. in addition to providing an overview of the framework, this chapter also discusses existing approaches for assessing software security and texas holdem poker rule.

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