a) Rock-Paper-Scissors-Dynamite: This game is like Rock-Paper-Scissors but with the addition of Dynamite which beats Rock and Paper but loses to Scissors (which cut the wick of the dynamite). Express this as a game (find A matrix) and derive equilibrium strategies. b) Weighted Rock-Paper-Scissors: This game is like Rock-Paper-Scissors but with weights, i.e., Rock beats Scissors by 5 points, Paper beats Rock by 3 points and Scissors beats Paper by 1 point. Each player wants to maximize their total points. Express this as a game (find A matrix) and derive equilibrium strategies.<\/p>\n\n\n\n
The Correct Answer and Explanation is :<\/strong><\/mark><\/p>\n\n\n\n
Solution for Rock-Paper-Scissors-Dynamite (RPSD)<\/strong><\/h3>\n\n\n\n
Game Matrix Representation<\/strong><\/h4>\n\n\n\n
We have four strategies: Rock (R), Paper (P), Scissors (S), and Dynamite (D)<\/strong>. The payoff matrix for Player 1 (where Player 2 receives the negative of each value) is:<\/p>\n\n\n\n
Rock (R)<\/strong> beats Scissors (+1), loses to Paper and Dynamite (-1).<\/li>\n\n\n\n
Paper (P)<\/strong> beats Rock (+1), loses to Scissors and Dynamite (-1).<\/li>\n\n\n\n
Scissors (S)<\/strong> beats Paper and Dynamite (+1), loses to Rock (-1).<\/li>\n\n\n\n
Dynamite (D)<\/strong> beats Rock and Paper (+1) but loses to Scissors (-1).<\/li>\n<\/ul>\n\n\n\n
Equilibrium Strategy<\/strong><\/h4>\n\n\n\n
To find a mixed-strategy Nash equilibrium<\/strong>, let Player 1 play R, P, S, and D<\/strong> with probabilities (x_1, x_2, x_3, x_4). Similarly, Player 2 plays with probabilities (y_1, y_2, y_3, y_4). The optimal strategy is to make the opponent indifferent among their choices, leading to:<\/p>\n\n\n\n
Rock (R)<\/strong> beats Scissors by 5 points<\/strong>, loses to Paper by 3 points<\/strong>.<\/li>\n\n\n\n
Paper (P)<\/strong> beats Rock by 3 points<\/strong>, loses to Scissors by 1 point<\/strong>.<\/li>\n\n\n\n
Scissors (S)<\/strong> beats Paper by 1 point<\/strong>, loses to Rock by 5 points<\/strong>.<\/li>\n<\/ul>\n\n\n\n
Equilibrium Strategy<\/strong><\/h4>\n\n\n\n
Let Player 1 play R, P, S<\/strong> with probabilities (p_1, p_2, p_3), and Player 2 plays with (q_1, q_2, q_3). The expected payoff must be equalized across choices:<\/p>\n\n\n\n
This means players should favor Paper the most<\/strong>, then Rock<\/strong>, and play Scissors the least<\/strong> due to the scoring differences.<\/p>\n\n\n\n\n\n\n\n
Conclusion<\/strong><\/h3>\n\n\n\n
\n
RPSD<\/strong> has a balanced 1\/3, 1\/3, 1\/6, 1\/6<\/strong> probability mix.<\/li>\n\n\n\n
WRPS<\/strong> favors Paper with 5\/9 probability, Rock with 3\/9, and Scissors with 1\/9.<\/li>\n<\/ul>\n\n\n\n
Now, I will generate an image illustrating the game matrices and strategies.<\/p>\n\n\n\n<\/figure>\n\n\n\n
Here is the generated image illustrating the game matrices and equilibrium strategies for Rock-Paper-Scissors-Dynamite (RPSD) and Weighted Rock-Paper-Scissors (WRPS). Let me know if you need any modifications or further clarifications!<\/p>\n","protected":false},"excerpt":{"rendered":"
a) Rock-Paper-Scissors-Dynamite: This game is like Rock-Paper-Scissors but with the addition of Dynamite which beats Rock and Paper but loses to Scissors (which cut the wick of the dynamite). Express this as a game (find A matrix) and derive equilibrium strategies. b) Weighted Rock-Paper-Scissors: This game is like Rock-Paper-Scissors but with weights, i.e., Rock beats […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-195445","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195445","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=195445"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195445\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195445"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195445"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195445"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
![\"\"](\"https:\/\/learnexams.com\/blog\/wp-content\/uploads\/2025\/02\/image-1637.png\")
<\/figure>\n\n\n\n