A simple game that has no solution

The fol­low­ing sim­ple game has one solu­tion that seems cor­rect, but isn’t. Can you figure out why?

The Game

Player One moves first. He must pick A, B, or C. If Player One picks A the game ends and Player Two does noth­ing. If Player One picks B or C, Player Two will be told that Player One picked B or C, but will not be told which of these two strate­gies Player One picked, Player Two must then pick X or Y, and then the game ends. The fol­low­ing shows the Play­ers’ pay­offs for each pos­si­ble out­come. Player One’s pay­off is listed first.

A 3,0 [And Player Two never got to move.]

B,X 2,0

B,Y 2,2

C,X 0,1

C,Y 6,0

The play­ers are ra­tio­nal, each player cares only about max­i­miz­ing his own pay­off, the play­ers can’t com­mu­ni­cate, they play the game only once, this game is all that will ever mat­ter to them, and all of this plus the pay­offs and the game struc­ture is com­mon knowl­edge.

Guess what will hap­pen. Imag­ine you are re­ally play­ing the game and de­cide what you would do as ei­ther Player One, or as Player Two if you have been told that you will get to move. To figure out what you would do you must for­mu­late a be­lief about what the other player has/​will do, and this will in part be based on your be­lief about his be­lief of what you have/​will do.

An In­cor­rect Ar­gu­ment for A

If Player One picks A he gets 3, whereas if he picks B he gets 2 re­gard­less of what Player Two does. Con­se­quently, Player One should never pick B. If Player One picks C he might get 0 or 6 so we can’t rule out Player One pick­ing C, at least with­out first figur­ing out what Player Two will do.

Player Two should as­sume that Player One will never pick B. Con­se­quently, if Player Two gets to move he should as­sume that C was played and there­fore Player Two should re­spond with X. If Player One be­lieves that Player Two will, if given the chance to move, pick X, then Player One is best off pick­ing A. In con­clu­sion, Player One will pick A and Player Two will never get to move.

Why the Game Has No Solution

I be­lieve that the above logic is wrong, and in­deed the game has no solu­tion. My rea­son­ing is given in rot13. (Copy what is be­low and paste at this link to con­vert to English.)

http://​​rot13.com/​​

Vs gur no­bir nanylfvf jrer pbeerpg Cyn­lre Gjb jb­hyq oryvrir ur jvyy ar­ire zbir. Fb jung unc­craf vs Cyn­lre Gjb qbrf trg gb zbir? Vs Cyn­lre Gjb trgf gb zbir jung fub­hyq uvf oryvrs or nobhg jung Cyn­lre Bar qvq tvira gung Cyn­lre Gjb xabjf Cyn­lre Bar qvq abg cvpx N? Cyn­lre Gjb pna’g nffhzr gung P jnf cyn­lrq. Vs vg jrer gehr gung vg’f pbzzba xab­jyrqtr gung Cyn­lre Bar jb­hyq ar­ire cynl O, gura vg fub­hyq or pbzzba xab­jyrqtr gung Cyn­lre Gjb jb­hyq ar­ire cynl L, ju­vpu jb­hyq zrna gung Cyn­lre Bar jb­hyq ar­ire cynl P, ohg pyr­neyl Cyn­lre Bar unf cvpxrq O be P fb fbzr­gu­vat vf je­bat.

Zber nof­genpgyl, vs V qrirybc n gurbel gung lbh jba’g gnxr npgvba Y, naq guvf arprffnevyl erfhygf va gur vz­cyvp­ngvba gung lbh jba’g qb npgvba Z, gura vs lbh unir pyr­neyl qbar rvgure Y be Z zl bevt­vany gurbel vf vainyvq. V’z abg nyy­b­jrq gb nffhzr gung lbh zhfg unir qbar Z whfg orpn­hfr zl vavgvny cebbs ubyq­vat gung lbh jba’g qb Y gbbx sr­jre fgrcf guna zl cebbs sbe jul lbh jba’g qb Z qvq.

Abar vs guvf jb­hyq or n ce­boyrz vs vg jrer veengvbany sbe Cyn­lre Bar gb abg cvpx N. Ns­gre nyy, V unir nffhzrq en­gvbanyvgl fb V’z abg nyy­b­jrq gb cbfghyngr gung Cyn­lre Bar jvyy qb fbzr­gu­vat veengvbany. Ohg vg’f veengvbany sbe Cyn­lre Bar gb Cvpx P bayl vs ur rfgvzn­grf gung gur ce­bonovyvgl bs Cyn­lre Gjb erfcbaq­vat jvgu L vf fhssvpvragyl ybj. Cyn­lre Gjb’f zbir jvyy qr­craq ba uvf oryvrsf bs jung Cyn­lre Bar unf qbar vs Cyn­lre Bar unf abg cvpxrq N. Pbafrdhragyl, jr pna bayl fnl vg vf veengvbany sbe Cyn­lre Bar gb abg cvpx N ns­gre jr unir svtherq bhg jung oryvrs Cyn­lre Gjb jb­hyq unir vs Cyn­lre Gjb trgf gb cynl. Naq guvf oryvrs bs Cyn­lre Gjb pna’g or on­frq ba gur nffhzcgvba gung Cyn­lre Bar jvyy ar­ire cvpx O orpn­hfr guvf erfhygf va Cyn­lre Gjb oryvri­vat gung Cyn­lre Bar jvyy ar­ire cvpx P rvgure, ohg pyr­neyl vs Cyn­lre Gjb trgf gb zbir rvgure O be P unf orra cvpxrq.

Va fhz, gb svaq n fby­hgvba sbe gur tnzr jr arrq gb xabj jung Cyn­lre Gjb jb­hyq qb vs ur trgf gb zbir, ohg gur bayl ern­fbanoyr pnaqvqngr fby­hgvba unf Cyn­lre Gjb ar­ire zbi­vat fb jr unir n pbagen­qvpgvba naq V unir ab vqrn jung gur ev­tug nafjre vf. Guvf vf n trareny ce­boyrz va tnzr gurbel ju­rer n fby­hgvba erd­hverf svthe­vat bhg jung n cyn­lre jb­hyq qb vs ur trgf gb zbir, ohg nyy gur ern­fbanoyr fby­hgvbaf unir guvf cyn­lre ar­ire zbi­vat.

Up­date: Emile has a great an­swer if you as­sume a “trem­bling hand.”