{"id":779922,"date":"2020-03-24T17:31:52","date_gmt":"2020-03-24T15:31:52","guid":{"rendered":"https:\/\/pokeriomokykla.com\/gto-in-poker-and-beyond-nash-equilibrium"},"modified":"2025-04-28T23:30:38","modified_gmt":"2025-04-28T21:30:38","slug":"gto-in-poker-and-beyond-nash-equilibrium","status":"publish","type":"post","link":"https:\/\/pokeriomokykla.com\/en\/about-gto-poker","title":{"rendered":"GTO in poker and beyond: Nash Equilibrium"},"content":{"rendered":"<figure class=\"wp-block-image aligncenter\"><img decoding=\"async\" width=\"1024\" height=\"576\" src=\"https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-pokeryje.png\" alt=\"Poker GTO\" class=\"wp-image-761411\" srcset=\"https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-pokeryje.png 1024w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-pokeryje-300x169.png 300w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-pokeryje-768px-432px.png 768w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-pokeryje-320px-180px.png 320w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-pokeryje-120px-68px.png 120w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-pokeryje-336px-188px.png 336w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-pokeryje-213px-120px.png 213w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-pokeryje-700px-394px.png 700w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><\/figure>\n<\/p>\n<p>Game theory belongs to the field of mathematics. Games in this context have a very broad meaning. Essentially, it is a theory analyzing any situations where rational agents (usually people) act according to a set (or predictable) strategy. From this definition alone, it is easy to determine the application of <a href=\"https:\/\/pokeriomokykla.com\/pokerio-teorija-kaip-turetumete-galvoti-apie-zaidima\" target=\"_blank\" rel=\"noopener\">GTO concepts in poker<\/a>.<\/p>\n<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_76 ez-toc-wrap-left counter-hierarchy ez-toc-counter ez-toc-custom ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title\" style=\"cursor:inherit\">Table of Contents<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg style=\"fill: #000000;color:#000000\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" class=\"list-377408\" width=\"20px\" height=\"20px\" viewBox=\"0 0 24 24\" fill=\"none\"><path d=\"M6 6H4v2h2V6zm14 0H8v2h12V6zM4 11h2v2H4v-2zm16 0H8v2h12v-2zM4 16h2v2H4v-2zm16 0H8v2h12v-2z\" fill=\"currentColor\"><\/path><\/svg><svg style=\"fill: #000000;color:#000000\" class=\"arrow-unsorted-368013\" xmlns=\"http:\/\/www.w3.org\/2000\/svg\" width=\"10px\" height=\"10px\" viewBox=\"0 0 24 24\" version=\"1.2\" baseProfile=\"tiny\"><path d=\"M18.2 9.3l-6.2-6.3-6.2 6.3c-.2.2-.3.4-.3.7s.1.5.3.7c.2.2.4.3.7.3h11c.3 0 .5-.1.7-.3.2-.2.3-.5.3-.7s-.1-.5-.3-.7zM5.8 14.7l6.2 6.3 6.2-6.3c.2-.2.3-.5.3-.7s-.1-.5-.3-.7c-.2-.2-.4-.3-.7-.3h-11c-.3 0-.5.1-.7.3-.2.2-.3.5-.3.7s.1.5.3.7z\"\/><\/svg><\/span><\/span><\/span><\/a><\/span><\/div>\n<nav><ul class='ez-toc-list ez-toc-list-level-1 eztoc-toggle-hide-by-default' ><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-1\" href=\"https:\/\/pokeriomokykla.com\/en\/about-gto-poker\/#TOP_poker_rooms\" >TOP poker rooms<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-2\" href=\"https:\/\/pokeriomokykla.com\/en\/about-gto-poker\/#GTO_in_poker_%E2%80%93_Nash_equilibrium\" >GTO in poker &#8211; Nash equilibrium<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-3\" href=\"https:\/\/pokeriomokykla.com\/en\/about-gto-poker\/#GTO_and_Nash_Equilibrium_myths\" >GTO and Nash Equilibrium myths<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-4\" href=\"https:\/\/pokeriomokykla.com\/en\/about-gto-poker\/#GTO_in_poker_%E2%80%93_Recalculating_Nash_equilibrium\" >GTO in poker &#8211; Recalculating Nash equilibrium<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-5\" href=\"https:\/\/pokeriomokykla.com\/en\/about-gto-poker\/#Best_Online_Poker_Rooms\" >Best Online Poker Rooms<\/a><\/li><\/ul><\/nav><\/div>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"TOP_poker_rooms\"><\/span>TOP poker rooms<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<div class='widget-card placeholder--card'\n     data-source-gql='https:\/\/scg-new-live.stellate.sh\/'\n     data-source-widgetjs='https:\/\/widget.affprojects.com\/widget.js'\n     data-shortcode='widget-card'\n     data-shortcode-text='[top-poker-rooms-for-post]'\n     data-props='{\n\"show_more_button\":true,\n\"show_more_button_text\":\"Show More Bookmakers\",\n\"show_more_text\":false,\n\"show_less_text\":false,\n\"vpn_notification\":false,\n\"show_more_items_amount\":null,\n\"scrollable\":\"always\",\n\"violet_bonus_text\":true,\n\"ab_testing_show_more_button_rework\":true\n}'\n     data-queries='{\n\"type\":\"TOP-Poker\",\n\"limit\":10,\n\"mobile_limit\":10,\n\"show_new\":1,\n\"button_text_type\":\"TOP\",\n\"language\":\"en\",\n\"project\":\"PM\",\n\"page\":1\n}'\n     ><\/div>\n\n<p>One of the most well-known and important principles in poker game theory is Nash equilibrium. Many poker players know this concept only from tables that are Push\/Fold All in preflop.<\/p>\n<p>These tables usually indicate at how many BB it is <i>possible<\/i> to shove or call All in against one player. Of course, such tables can be created for a larger number of players, but it is usually not as easy (and useful).<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"GTO_in_poker_%E2%80%93_Nash_equilibrium\"><\/span>GTO in poker &#8211; Nash equilibrium<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>Nash equilibrium first appeared in poker in the form of well-known \u201c<a href=\"https:\/\/www.holdemresources.net\/hune\" target=\"_blank\" rel=\"nofollow noopener\">preflop All in<\/a>\u201d tables, as it is the easiest state to calculate in the game. The fewer chips and players remain, the fewer possible decisions (theoretically) each of them can implement.<\/p>\n<p>To better understand how the calculation of these situations arose, it is necessary to understand what state is considered Nash equilibrium. Nash equilibrium is a state of the game that is reached when the following conditions exist:<\/p>\n<ol class=\"wp-block-list\">\n<li>There are at least two players.<\/li>\n<li>A non-cooperative game is played (one where players can win something at the expense of others).<\/li>\n<li>The strategies of all players are known.<\/li>\n<li>Strategies are formed such that no player can win more by changing only their strategy.<\/li>\n<\/ol>\n<p>The fourth point is the most important in this case. Nash equilibrium describes a strategy where all players play optimally. In poker terms, it is a plan of action such that changing only it cannot win more EV.<\/p>\n<p>In poker, Nash Equilibrium would mean 0 EV for all players.<\/p>\n<figure class=\"wp-block-image aligncenter GTO example not in poker wp-image-761414 size-full\"><img decoding=\"async\" width=\"938\" height=\"405\" src=\"https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/GTO-pavyzdys.png\" alt=\"GTO example not in poker\" class=\"wp-image-761414\" srcset=\"https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/GTO-pavyzdys.png 938w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/GTO-pavyzdys-300x130.png 300w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/GTO-pavyzdys-768px-332px.png 768w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/GTO-pavyzdys-278px-120px.png 278w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/GTO-pavyzdys-700px-302px.png 700w\" sizes=\"(max-width: 938px) 100vw, 938px\" \/><figcaption class=\"wp-element-caption\">Prisoner's dilemma &#8211; the most common example of game theory<\/figcaption><\/figure>\n<p><a href=\"https:\/\/www.britannica.com\/biography\/John-Nash\" target=\"_blank\" rel=\"noopener\">John Forbes Nash<\/a>, the discoverer of this equilibrium, has proven a very important phenomenon &#8211; the theory of the existence of Nash equilibrium. This theorem states that Nash Equilibrium can be found in all games where the number of decisions is not infinite and winnings correlate with players' actions (or provide an incentive to create a strategy based on certain principles).<\/p>\n<p>This means that Nash Equilibrium exists not only in tables but in the entire poker game from preflop to the final river decision. For every poker strategy with at least one rule, there is a GTO solution that would manifest as a unilateral Nash equilibrium. We will discuss the search for such Game Theory Optimal poker solutions in the next part of this article.<\/p>\n<p>Such strategies can already be found with the offered solvers. So, GTO is not just for \u201ctop top regs.\u201d<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"GTO_and_Nash_Equilibrium_myths\"><\/span>GTO and Nash Equilibrium myths<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>As mentioned earlier, we often hear about Nash Equilibrium in poker only when talking about preflop tables, and GTO only when discussing solvers or as some very distant and abstract strategy. In fact, these concepts apply to all levels of poker.<\/p>\n<p>Why were Nash tables created first before starting to create and talk about more complex GTO strategies? First, as is well known, these tables are best applied when there is an extremely low effective stack size.<\/p>\n<p>The fewer chips, the fewer possibilities both pre and postflop. This reason is compounded by the importance of postflop when deep. Many players do not like to open shove AA at every opportunity.<\/p>\n<p>Secondly, a much more important reason is the simplification of the game. Nash equilibrium requires understanding at least one player's strategy <i>fully<\/i>, which in many poker situations is not realistic. Many recreational players probably only have the beginnings of a strategy instead of rational game rules. For this reason, achieving <a href=\"https:\/\/pokeriomokykla.com\/cash-games-pradmenys-pokerio-mokykloje\" target=\"_blank\" rel=\"noopener\">Nash equilibrium in poker<\/a> was only possible by simplifying the game.<\/p>\n<p>GTO solvers and other modern programs allow calculating a strategy that approaches Nash equilibrium. The best solvers, however, still do not achieve perfection, but for many players, this is not important. The average exploitability per pot does not reach 0.1%, so in poker practice, it means nothing, as players will not be able to memorize all possible GTO strategies.<\/p>\n<p>The most important practical and theoretical aspect of GTO and solvers is the formulation of a perfect counter-strategy. For any poker strategy, an optimal, maximum EV strategy can be found. This strategy will be maximally profitable until the other player changes it.<\/p>\n<figure class=\"wp-block-image aligncenter Poker GTO solver example wp-image-761413 size-full\"><img decoding=\"async\" width=\"1024\" height=\"656\" src=\"https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-Solveris.jpg\" alt=\"Poker GTO solver example\" class=\"wp-image-761413\" srcset=\"https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-Solveris.jpg 1024w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-Solveris-300x192.jpg 300w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-Solveris-768px-492px.jpg 768w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-Solveris-187px-120px.jpg 187w, https:\/\/pokeriomokykla.com\/app\/uploads\/2020\/03\/Gto-Solveris-700px-448px.jpg 700w\" sizes=\"(max-width: 1024px) 100vw, 1024px\" \/><figcaption class=\"wp-element-caption\">Poker GTO solver example<\/figcaption><\/figure>\n<p>Of course, to start even the simplest optimal strategies, it is necessary to know about the opponent's playing style. It would be best to know specific rules, as these can greatly affect the final result.<\/p>\n<p>For example, if we had two players, one who in position 100% flops top pair c-bets \u00bd pot size, and another who in position bets only 50% flops cbet (with better kickers), even if all their other game rules are identical, the optimal strategies will differ significantly.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"GTO_in_poker_%E2%80%93_Recalculating_Nash_equilibrium\"><\/span>GTO in poker &#8211; Recalculating Nash equilibrium<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>One of the most important statements added to all these Nash tables is that it is a simplified poker game. In it, the Small Blind can only perform All in or Fold, and the Big Blind can only Call or Fold.<\/p>\n<p>Players cannot perform any other actions such as Raise and can never go to postflop. This is the main reason why these tables always show Small Blind All in 20+ BB or even higher decisions, although they are usually not applicable in practice.<\/p>\n<p>Finally, many of these tables and decisions are created based on the calculations from the book Mathematics of Poker. Contrary to what many players think, we should not blindly apply these tables to our game, as we, especially against recreational players, will never achieve the desired Nash Equilibrium with this table.<\/p>\n<p>A very simple but well-illustrated example is a player who shoves any two cards 25 BB deep. In this case, the tables would be neither close to Nash Equilibrium nor close to maximum EV, as the optimal strategy would be different.<\/p>\n<p>An optimal strategy would allow for a wider call, as the Small Blind would shoot significantly more poor hands than presented in the solutions.<\/p>\n<h3 class=\"wp-block-heading\">Changes in Nash Equilibrium<\/h3>\n<p>Of course, in reality, there will be many more minor deviations from the strategy described in the tables. Spin n Go and HUSNG players know that in the 14-9 BB range, Heads Up will be played very differently. Some opponents will limp a lot, some will have more minraises in their ranges, and so on.<\/p>\n<p>Due to these differences, the optimal shove\/fold range will always differ quite significantly, so such situations should always be reviewed in ICMizer, especially from the BB perspective. Of course, in situations where a player is encountered only once, intuition or other simplifications of the situation may have to be used.<\/p>\n<figure class=\"wp-block-embed is-type-video is-provider-youtube wp-block-embed-youtube wp-embed-aspect-4-3 wp-has-aspect-ratio\">\n<div class=\"wp-block-embed__wrapper\">\n<iframe title=\"Comparing Nash Heads Up Ranges Versus The Population\" width=\"640\" height=\"480\" src=\"https:\/\/www.youtube.com\/embed\/xnBlnLLq8Ks?feature=oembed\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share\" referrerpolicy=\"strict-origin-when-cross-origin\" allowfullscreen><\/iframe>\n<\/div>\n<\/figure>\n<p>This is a very old video about Nash changes, but it perfectly illustrates how optimal play should change.<\/p>\n<p>Relying on push\/fold tables is possible as the game approaches 1 BB effective stacks, as all possible strategies approach Nash Equilibrium. Except for the &#8220;fold everything&#8221; strategy, many strategies at 2, 3, 4, or 5 big blinds will be almost identical or differ very slightly.<\/p>\n<p>Thus, in this article, we discussed the basics of GTO poker and the most well-known concept of this theory, Nash Equilibrium. In other articles, more attention will be given to the search for optimal strategies and the principles of solver operation.<\/p>\n<h2 class=\"wp-block-heading\"><span class=\"ez-toc-section\" id=\"Best_Online_Poker_Rooms\"><\/span>Best Online Poker Rooms<span class=\"ez-toc-section-end\"><\/span><\/h2>\n\n<div class='widget-card placeholder--card'\n     data-source-gql='https:\/\/scg-new-live.stellate.sh\/'\n     data-source-widgetjs='https:\/\/widget.affprojects.com\/widget.js'\n     data-shortcode='widget-card'\n     data-shortcode-text='[top-poker-rooms-for-post]'\n     data-props='{\n\"show_more_button\":true,\n\"show_more_button_text\":\"Show More Bookmakers\",\n\"show_more_text\":false,\n\"show_less_text\":false,\n\"vpn_notification\":false,\n\"show_more_items_amount\":null,\n\"scrollable\":\"always\",\n\"violet_bonus_text\":true,\n\"ab_testing_show_more_button_rework\":true\n}'\n     data-queries='{\n\"type\":\"TOP-Poker\",\n\"limit\":10,\n\"mobile_limit\":10,\n\"show_new\":1,\n\"button_text_type\":\"TOP\",\n\"language\":\"en\",\n\"project\":\"PM\",\n\"page\":1\n}'\n     ><\/div>\n\n","protected":false},"excerpt":{"rendered":"<p>Game theory belongs to the field of mathematics. Games in&#8230;<\/p>\n","protected":false},"author":4481,"featured_media":776205,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"footnotes":""},"categories":[1658],"tags":[],"class_list":["post-779922","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-poker-strategy"],"acf":[],"_links":{"self":[{"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/posts\/779922","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/users\/4481"}],"replies":[{"embeddable":true,"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/comments?post=779922"}],"version-history":[{"count":2,"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/posts\/779922\/revisions"}],"predecessor-version":[{"id":1094261,"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/posts\/779922\/revisions\/1094261"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/media\/776205"}],"wp:attachment":[{"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/media?parent=779922"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/categories?post=779922"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/pokeriomokykla.com\/en\/wp-json\/wp\/v2\/tags?post=779922"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}