[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-572927-105":3,"detail-sidebar-cat-1-en-105":80,"doc-detail-572927-en":126},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","ultimate-x-poker-analysis","Ultimate X Poker Analysis","","Ultimate X Poker Analysis studies video poker strategy for casinos where machine outcome probabilities are known, enabling determination of optimal play. Because optimal decisions depend not only on the current hand but also on how the chosen action influences the subsequent hand, the work formulates the problem as a large non-discounted Markov Decision process. The paper develops the expected return framework, discusses outcome valuation and subset selection, and explains how optimal actions and strategies can be computed despite the computational complexity.",{"@graph":14,"@context":72},[15,34,55],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/template/","Template",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/template/presentations/","Presentations",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/template/ultimate-x-poker-analysis/572927/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/ultimate-x-poker-analysis/572927.png","ImageObject",442,249,{"name":42,"@type":43},"Cart","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-19","2026-10-13",true,{"@type":52,"interactionType":53,"userInteractionCount":33},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"Why does Ultimate X pose a harder analysis challenge than many other video poker games?","Question",{"text":62,"@type":63},"Optimal play depends on both the current hand and the impact on a subsequent hand, which creates a large non-discounted Markov Decision problem.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How is an expected return for choosing a subset of cards defined?",{"text":67,"@type":63},"For a given hand and subset, the expected return sums outcome values weighted by the probabilities of outcomes computed over all possible completions of the subset.",{"name":69,"@type":60,"acceptedAnswer":70},"What makes a brute-force computation impractical in standard 52-card video poker?",{"text":71,"@type":63},"There are about 2,598,960 possible hands (ignoring order) and many subset selection possibilities, so evaluating every hand and all resulting actions would require examining an enormous number of combinations.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},572927,1792172014,{"code":4,"msg":81,"data":82},"success",[83,87,92,97,102,107,112,117,122],{"id":84,"doc_module":22,"doc_module_name":25,"category_name":29,"show_sort_weight":85,"slug":86},11,90,"presentations",{"id":88,"doc_module":22,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},12,"Resumes",80,"resumes",{"id":93,"doc_module":22,"doc_module_name":25,"category_name":94,"show_sort_weight":95,"slug":96},14,"Invoices",70,"invoices",{"id":98,"doc_module":22,"doc_module_name":25,"category_name":99,"show_sort_weight":100,"slug":101},15,"Posters",60,"posters",{"id":103,"doc_module":22,"doc_module_name":25,"category_name":104,"show_sort_weight":105,"slug":106},16,"Social Media",50,"social-media",{"id":108,"doc_module":22,"doc_module_name":25,"category_name":109,"show_sort_weight":110,"slug":111},17,"Forms",40,"forms",{"id":113,"doc_module":22,"doc_module_name":25,"category_name":114,"show_sort_weight":115,"slug":116},18,"Letters",30,"letters",{"id":118,"doc_module":22,"doc_module_name":25,"category_name":119,"show_sort_weight":120,"slug":121},21,"Paper Templates",5,"papers-templates",{"id":123,"doc_module":22,"doc_module_name":25,"category_name":124,"show_sort_weight":4,"slug":125},158,"General","general-158",{"code":4,"msg":81,"data":127},{"doc_id":78,"user_id":128,"nickname":42,"user_avatar":129,"doc_module":22,"category_id":84,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":130,"file_id":131,"file_url":132,"file_type":133,"file_size":134,"view_count":33,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":113,"language":135,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":136,"faqs":137,"seo_title":138,"seo_description":12,"update_tm":139,"read_time":140},18829141979164,"https://eur-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","Ultimate X Poker Analysis  \nGary J. Koehler  \nJohn B. Higdon Eminent Scholar, Emeritus  \nDepartment of Information Systems and Operations Management, 351 BUS, The Warrington College of Business, University of Florida, Gainesville, FL 32611, ([koehler@ufl.edu](koehler@ufl.edu)).  \nCasinos offer many forms of video poker machines (technically, slot machines) . Finding optimal strategies to play these games is tedious but straightforward. However, a recently introduced game poses a greater analysis challenge since optimal play depends on not only the current hand of play but the impact on a subsequent hand. This leads to analyzing a very large non-discounted Markov Decision problem. This paper explores this analysis.  \nKey words: Gambling, non-discounted Markov Decision Problem, poker.  \nAugust 4, 2010  \nRevised March, 2011  \nMinor Clean-up, October, 2015 New Section on Variance, December, 2015  \nCopyright © 2010, 2011 Gary J. Koehler  \n1. Introduction  \nMost casinos offer various types of video poker slot machines, such as Jacks or Better or Deuces Wild. Unlike most other types of slot machines, the probabilities of outcomes are known to players and optimal play can be determined. After a bet is placed, the machine displays five cards randomly drawn without replacement from a deck of cards. The player selects zero, one or more of these to hold, discarding the rest. The machine then fills-in the discarded cards drawn randomly from the deck with the initial five cards removed. The player wins an amount based on the outcome of this completed hand and on the pay-table for the selected game. A typical table of payouts per betting unit (e.g., a quarter or dollar) for placing a maximum bet (typically of five betting units) is as shown in Table 1 :  \n\n| Jacks or Better |  |  | Deuces Wild |  |\n| --- | --- | --- | --- | --- |\n| Outcome | Value |  | Outcome | Value |\n| Royal Straight Flush | 800 |  | Natural Royal Straight Flush | 800 |\n| Straight Flush | 50 |  | Four Deuces | 200 |\n| Four of a Kind | 25 |  | Wild Royal Straight Flush | 25 |\n| Full House | 9 |  | Five of a Kind | 15 |\n| Flush | 6 |  | Straight Flush | 9 |\n| Straight | 4 |  | Four of a Kind | 4 |\n| Three of a Kind | 3 |  | Full House | 4 |\n| Two Pair | 2 |  | Flush | 3 |\n| Jacks or Better Pair | 1 |  | Straight | 2 |\n|  |  |  | Three of a Kind | 1 |\n| Otherwise | 0 |  | Otherwise | 0 |\n\nTable 1: Outcome Values per betting unit (on a maximum bet) .  \nThese games have well-known optimal strategies (e.g., see [http://wizardofodds.com/videopoker](http://wizardofodds.com/videopoker)[ ](http://wizardofodds.com/videopoker)[Shackleford 2010](Shackleford 2010)) .  \nFinding an optimal strategy is straightforward, though tedious. Let 􀀕 be the set of all five-card hands that can be dealt from a deck of cards. In most video poker games, the deck is a standard one containing 52 cards. (Some games, like Joker Poker, include additional cards-in Joker Poker there is one additional card, a Joker, a “wild” card meaning it can be used as any card) . Also, in most games, the order of the cards in a five card hand is not important. However, in some variations, order is important. 􀀕 reflects these considerations once a particular game is selected.  \nFor any five card hand, H ∈ 􀀕 , let Hi be the ith subset of H , i=0,…,31 . For example, if H = {2H, JC, QD,3S, 7S } , meaning a hand containing a 2 of Hearts, a Jack of Clubs, a Queen of Diamonds, a 3 of Spades and a 7 of Spades, then  \nH0 = { }  \nH1 = {2H}  \nH2 = {JC }  \nH3 = {2H, JC }  \nH4 = {QD }  \nH5 = {2H, QD }  \nH6 = {JD, QD }  \nH7 = {2H, JD, QD }  \n...  \nH31 = {2H, JC, QD,3S, 7S }  \nLet Vj be the value of outcome j relevant to the game (e.g., a Flush) and Pj ( Hi ) = prob ( outcomej | Hi )  \nbe the probability of outcome j computed from all possible completions of subset i from the deck of cards with the cards listed in H removed. The expected return for choosing subset i is  \nRHi = Vj Pj ( Hi ) . The set of optimal actions, SH , for hand H ∈ ","cbCaiq4eSdLfwCy0","https://ap.wps.com/l/cbCaiq4eSdLfwCy0","pdf",271347,"English","# Introduction\n## Video poker game mechanics\n## Expected return and optimal actions\n## Strategy valuation and variance considerations\n## Computational complexity","[{\"question\":\"Why does Ultimate X pose a harder analysis challenge than many other video poker games?\",\"answer\":\"Optimal play depends on both the current hand and the impact on a subsequent hand, which creates a large non-discounted Markov Decision problem.\"},{\"question\":\"How is an expected return for choosing a subset of cards defined?\",\"answer\":\"For a given hand and subset, the expected return sums outcome values weighted by the probabilities of outcomes computed over all possible completions of the subset.\"},{\"question\":\"What makes a brute-force computation impractical in standard 52-card video poker?\",\"answer\":\"There are about 2,598,960 possible hands (ignoring order) and many subset selection possibilities, so evaluating every hand and all resulting actions would require examining an enormous number of combinations.\"}]","Ultimate X Poker Analysis | PDF",1791930130,6]