# Felix and Oscar are playing a simplified version of poker. Each makes an initial bet of 8 dollars…

Felix and Oscar are playing a simplified version of
poker. Each makes an initial bet of 8 dollars. Then each separately draws a
card, which may be High or Low with equal probabilities. Each sees his own card
but not that of the other. Then Felix decides whether to Pass or to Raise (bet
an additional 4 dollars). If he chooses to pass, the two cards are revealed and
compared. If the outcomes are different, the one who has the High card collects
the whole pot. The pot has 16 dollars, of which the winner himself contributed
8, so his winnings are 8 dollars. The loser’s payoff is 28 dollars. If the
outcomes are the same, the pot is split equally and each gets his 8 dollars
back (payoff 0). If Felix chooses Raise, then Oscar has to decide whether to
Fold (concede) or See (match with his own additional 4 dollars). If Oscar
chooses Fold, then Felix collects the pot irrespective of the cards. If Oscar
chooses See, then the cards are revealed and compared. The procedure is the
same as that in the preceding paragraph, but the pot is now bigger.

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(a) Show the game in extensive form. (Be careful about
information sets.) If the game is instead written in the normal form, Felix has
four strategies: (1) Pass always (PP for short), (2) Raise always (RR), (3)
Raise if his own card is High and Pass if it is Low (RP), and (4) the other way
round (PR). Similarly, Oscar has four strategies: (1) Fold always (FF), (2) See
always (SS), (3) See if his own card is High and Fold if it is Low (SF), and
(4) the other way round (FS).

(b) Show that the table of payoffs to Felix is as
follows:

(In each case, you will have to take an expected value by
averaging over the consequences for each of the four possible combinations of
the card draws.)

(c) Eliminate dominated strategies as far as possible.
Find the mixedstrategy equilibrium in the remaining table and the expected
payoff to Felix in the equilibrium.

(d) Use your knowledge of the theory of signaling and
screening to explain intuitively why the equilibrium has mixed strategies.