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This code accompanies the paper \[OpenCurlyDoubleQuote]when to hold \
\[OpenCurlyQuote]em\[CloseCurlyDoubleQuote] by Kaity Parsons, Peter Tingley \
and Emma Zajdela. \
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We first show the calcualtion from section 5. These can be done by hand \
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This solves the indifference equations for a general bet size a.\
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This substitutes a couple epcific bet sizes. You can change the 1 or 2 to any \
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P is the general payout function, in term of the cutoffs and the bet sizes. \
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\[OpenCurlyDoubleQuote]Payout\[CloseCurlyDoubleQuote] below is the payout at \
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This substitures to find the payout at a = 1, a = 2. Change the 1 or 2 to any \
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We now show calculations refered to in section 6. We also go a little \
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This solves the indifference equations for equilibrium L (so, in the case b \
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This plot shows that P1\[CloseCurlyQuote]s strategy
B with hand h<=x1 or with hand x3<=h<=x4
PC with hand x2<=h<=x3 or hand x4<=h<=1,
PF with hand x1<=h<=x2
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This defines functions giving P2\[CloseCurlyQuote]s expected payout as a \
function of their hand if they commit to using each of the possible \
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FP = fold if P1 bets, and pass if P1 passes.
FB = fold if P1 bets, and bet if P1 passes.
CP = call if P1 bets, and pass if P1 passes.
CB = call if P1 bets, and bet if P1 passes. \
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Cell["\<\
This plot shows that P2' s strategy
FB with hand h <= y1
FP with hand y1 <= h <= y2
CP with hand y2 <= h <= y3
CB with hand y3<=h<=1
has the property that, or any hand, they are using one of the behaviors with \
a maximal possible payout.\
\>", "Text",
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Cell["\<\
These all are clearly positive except for D3 and D7 (this can also be shown \
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an artifact, as shown by the following graph with restricted domain: \
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It can happen that D7 is negative, so y1 < x1, and that case is handled \
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This solves the indifference equations for equilibrium R, so in the case a >= \
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This lets you choose bet sizes and dispalys the corresponding cutoffs. Make \
sure a >= b or else the calculation will give an incorrect answer. Some \
values appear to give problems for technical reasons. For instance, setting \
a=1,b=2 gives an error because some denominators are zero, by if you \
substitue before solving for the cutoffs there is no problem.\
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This defines functions that give P1' s expected payout as a function of their \
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PC corresponds to passing then, if P2 bets, calling.
B corresponds to betting. Since P2 cannot raise, there is no more choice. \
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Cell["\<\
This plot shows that P1' s strategy
B with hand h <= x1 or with hand x3 <= h
PF with hand x1 <= h <= x2
PC with hand x1 <= h <= x3
has the property that, or any hand, they are using one of the behaviors with \
a maximal possible payout. \
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This defines functions that give P2\[CloseCurlyQuote]s expected payout as a \
function of their hand if they commit to using each of the possible \
behaviors:
FP = fold if P1 bets, and pass if P1 passes.
FB = fold if P1 bets, and bet if P1 passes.
CP = call if P1 bets, and pass if P1 passes.
CB = call if P1 bets, and bet if P1 passes.
Note that these function are different from in the b>=a case. \
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Cell["\<\
This plot shows that P2' s strategy
FB with hand h <= y1
FP with hand y1 <= h <= y2
CP with hand y2 <= h <= y3
CB with hand y3 <= h <= 1
has the property that, or any hand, they are using one of the behaviors with \
a maximal possible payout.\
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Cell["\<\
These are all clearly positive except D6. Note that the relative positions of \
x2, y2 and of y3,y2 are free; these do not change the equations so, although \
the pictures look a little different, we don\[CloseCurlyQuote]t consider \
these different cases. There is however a difference case for when D6 is \
negative. \
\>", "Text",
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Cell["\<\
As above, the two types of Nash equilibrium (L) and (R) do not cover all \
cases: when a and b are too far appart, it can happen that at equilibrium P1 \
bluffs with better hands than P2 bluffs with, and the equations. We now solve \
for when that happens, and find the corresponding equilibria.\
\>", "Text",
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Cell["\<\
We first consider L (b>a). We need to solve for where P1 starts wanting to \
bluff more often than P2. If a is similar to b then, by direct calculation, \
P1 bluffs less often than P2. Hence if there is a problematic region there \
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