{"id":2055,"date":"2010-09-22T16:08:26","date_gmt":"2010-09-22T20:08:26","guid":{"rendered":"http:\/\/www.noahgreenstein.com\/wordpress\/?p=2055"},"modified":"2010-09-22T16:08:26","modified_gmt":"2010-09-22T20:08:26","slug":"rock-paper-scissors","status":"publish","type":"post","link":"https:\/\/www.noahgreenstein.com\/wordpress\/2010\/09\/22\/rock-paper-scissors\/","title":{"rendered":"Rock Paper Scissors"},"content":{"rendered":"

Rock Paper Scissors is a game in which 2 players each choose one of three options: either rock, paper or scissors.\u00a0 Then the players simultaneously reveal their choices.\u00a0 Rock beats scissors but loses to paper (rock smashes scissors); Paper beats rock and loses to scissors (paper covers rock); Scissors beats paper but loses to rock (scissors cut paper).\u00a0 This cyclical payoff scheme (Rock > Scissors, Scissors > Paper, Paper > Rock) can be represented by this rubric:<\/p>\n\n\n\n\n\n\n\n
<\/td>\n<\/td>\nChild 2<\/td>\n<\/tr>\n
<\/td>\n<\/td>\nrock<\/td>\npaper<\/td>\nscissors<\/td>\n<\/tr>\n
Child 1<\/td>\nrock<\/td>\n0,0<\/td>\n-1,1<\/td>\n1,-1<\/td>\n<\/tr>\n
paper<\/td>\n1,-1<\/td>\n0,0<\/td>\n-1,1<\/td>\n<\/tr>\n
scissors<\/td>\n-1,1<\/td>\n1,-1<\/td>\n0,0<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n
.
\n<\/address>\n
(ref: Shor, Mikhael, “Rock Paper Scissors,” Dictionary of Game Theory Terms, Game Theory .net,\u00a0 <http:\/\/www.gametheory.net\/dictionary\/Games\/RockPaperScissors.html<\/a>>\u00a0 Web accessed: 22 September 2010)<\/address>\n

However, if we want to describe the game of Rock Paper Scissors – not just the payoff scheme – how are we to do it?<\/p>\n

Ordinary logics have no mechanism for representing simultaneous play.\u00a0 Therefore Rock Paper Scissors is problematic because there is no way to codify the simultaneous revelation of the players’ choices.<\/p>\n

However, let’s treat the simultaneous revelation of the players’ choices as a device to prevent one player from knowing the choice of the other.\u00a0 If one player were to know the choice of the other, then that player would always have a winning strategy by selecting the option that beats the opponent’s selection.\u00a0 For example, if Player 1 knew (with absolute certainty) that Player 2 was going to play rock, then Player 1 would play paper, and similarly for the other options.\u00a0 Since certain knowledge of the opponent’s play trivializes and ruins the game, it is this knowledge that must be prevented.<\/p>\n

Knowledge – or lack thereof – of moves can be represented within certain logics.\u00a0 Ordinarily all previous moves within logic are known, but if we declare certain moves to be independent from others, then those moves can be treated as unknown.\u00a0 This can be done in Independence Friendly Logic<\/a>, which allows for explicit dependence relations to be stated.<\/p>\n

So, let’s assume our 2 players, Abelard (\u2200) and Eloise (\u2203) each decide which of the three options he or she will play out of the Domain {r, p, s} .\u00a0 These decisions are made without knowledge of what the other has chosen, i.e. independently of each other.<\/p>\n

\u2200x \u2203y\/\u2200x<\/p>\n

This means that Abelard chooses a value for x first and then Eloise chooses a value for y.\u00a0 The \/\u2200x next to y means that the choice of y is made independently from, without knowledge of the value of, x.<\/p>\n

R-P-S:<\/strong> \u2200x \u2203y\/\u2200x (Vxy)<\/p>\n

The decisions are then evaluated according to V, which is some encoding of the above rubric like this:<\/p>\n\n\n\n\n\n\n\n\n\n
V:<\/strong><\/td>\nx=y \u2192 R-P-S<\/td>\n&<\/td>\n<\/tr>\n
<\/td>\nx=r & y=s \u2192 T<\/td>\n&<\/td>\n<\/tr>\n
x=r & y=p \u2192 F<\/td>\n&<\/td>\n<\/tr>\n
x=p & y=r \u2192 T<\/td>\n&<\/td>\n<\/tr>\n
x=p & y=s \u2192 F<\/td>\n&<\/td>\n<\/tr>\n
x=s & y=p \u2192 T<\/td>\n&<\/td>\n<\/tr>\n
x=s & y=r \u2192 F<\/td>\n<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

T means Abelard wins; F means Eloise wins.\u00a0 R-P-S means play more Rock Paper Scissors!<\/p>\n

—<\/p>\n

Johan van Benthem, Sujata Ghosh and Fenrong Liu put together a sophisticated and generalized logic for concurrent action:
\nhttp:\/\/www.illc.uva.nl\/Publications\/ResearchReports\/PP-2007-26.text-Jun-2007.pdf<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

Rock Paper Scissors is a game in which 2 players each choose one of three options: either rock, paper or scissors.\u00a0 Then the players simultaneously reveal their choices.\u00a0 Rock beats scissors but loses to paper (rock smashes scissors); Paper beats rock and loses to scissors (paper covers rock); Scissors beats paper but loses to rock (scissors cut paper).\u00a0 This cyclical payoff scheme (Rock > Scissors, Scissors > Paper, Paper > Rock) can be represented by […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[11,13,16,26],"tags":[250,251,256,268],"class_list":["post-2055","post","type-post","status-publish","format-standard","hentry","category-game-theory","category-independence-friendly-logic","category-logic","category-philosophy","tag-game-theory","tag-independence-friendly-logic","tag-logic","tag-philosophy"],"_links":{"self":[{"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/posts\/2055","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/comments?post=2055"}],"version-history":[{"count":0,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/posts\/2055\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/media?parent=2055"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/categories?post=2055"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.noahgreenstein.com\/wordpress\/wp-json\/wp\/v2\/tags?post=2055"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}