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I'm doing a self-study of Game Theory Evolving by Gintis, and am stuck on problem 4.16 "Poker with Bluffing".

The first question asks "Show that Ollie has 64 pure strategies and Stan has 8 pure strategies.". But no matter how I try to approach this, I can't get more then 4 strategies for Stan!

Here is the game tree for the game in question: enter image description here

The question marks (?) in the figure mean that the payoff depends on who has the higher card.

Here is the game description:

  • Two players, each with a deck of three cards: H (high), M(medium) or L (low).
  • Each puts \$1 in the pot, chooses random card
  • Ollie (P1) either stays or raises
  • Stan (P2) simultaneously also stays or raises
  • If both raise/stay - highest card wins the pot (tie - they take their money back)
  • If Ollie raises, Stan stays Ollie gets the \$3 pot.
  • If Stan raises and Ollie stays - Ollie gets another chance:
    -> Drop - Stand wins the \$3 pot
    -> Call - add \$1 to the pot.

Why does Stand have 8 pure strategies?

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This question came from our site for theoretical computer scientists and researchers in related fields.

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Stan has two pure strategies for each type of card. Since there are three types of cards (H, M, L), in total he has $2\times 2\times 2 = 8$ strategies. Here is a list of them:

  • Always stay.
  • Always raise.
  • Stay only if H.
  • Stay only if M.
  • Stay only if L.
  • Raise only if H.
  • Raise only if M.
  • Raise only if L.
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  • $\begingroup$ Ah!!! Thank you. So I could write them down as: SSS, RRR, SRR, RSR, RRS, RSS, SRS, SSR - where 1nd,2nd and 3rd position are actions for H, M, L cards $\endgroup$ – Andriy Drozdyuk Mar 27 '13 at 14:15
  • $\begingroup$ That's correct. $\endgroup$ – Yuval Filmus Mar 27 '13 at 17:49

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