Equations Used in the Section About the
Mixed Strategy for the War of Attrition, by Number

eq. 1:

eq. #2: E(fix(x=m), mix)

(Lifetime Net Benefits to Focal Strategy in Wins) - (Lifetime Costs to Focal Strategy in Losses)

 eq. #3a: Net Benefit to fix(x=m) in a Win = (V - x)

 eq. 3b: Net Benefit = Sum{(V - x) * (Prob of facing x)}

eq. #3c

 eq. 4a: Cost to fix(x=m) of Loss = (- m)

eq. #4b:

eq. #5:

eq. #6:

eq. #7:

eq. #8a:

eq. #8b:

eq. #9:

eq. #10: Calculation of deltaP(x)

eq. #11: Chance of Continuing per unit cost = exp(- 1/V) = 1/exp(1/V)

eq. 12: E(mix,fix(x)) - E(fix(x),fix(x)) > 0

eq. 13 

 eq. #14a:

which is the same as:

eq. 15: B for fix(x) vs.fix(x) =0.5 * (V - m) =0.5 * (V - x)

eq. 16a:

eq. 16b:

eq. #17: Cost paid by a fix(x) in losing to a fix(x): = 0.5 * x = 0.5 * m

eq. 18 E(fix(x=m), fix(x=m)) = 0.5*V - m

eq. 19a:

and the result of solving eq. 19a is:

eq. 19b:

eq. 20:

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