Critical Odds, Critical Probabilities

Suppose you must choose between action A and Action B when the probability of state S is p and the probability of state T is 1-p.
 
State S
Pr(S)=p		 State T
Pr(T)=1-p		 Expected
Utility
Action A AS AT p*AS +(1-p)*AT
Action B BS BT p*BS +(1-p)*BT

Without loss of generality, assume outcome AS is better than BS,
the only nontrivial case is then when outcome BT is better than AT.
Action A better than Action B

p*AS +(1-p)*AT > p*BS +(1-p)*BT
p*AS -p*BS > (1-p)*BT -(1-p)*AT
p*(AS-BS) > (1-p)*(BT-AT)

  p   > BT - AT
1 - p    AS - BS

 Action B better than Action A

pAS +(1-p)AT < pBS +(1-p)BT
PAS -pBS < (1-p)BT -(1-p)AT
p(AS-BS) < (1-p)(BT-AT)

  p   <BT - AT
1 - p   AS - BS

CALCULATING CRITICAL ODDS

Let W = BT - AT = advantage of B given T =   regret of A given T
AS - BS advantage of A given S  regret of B given S

W= "Critical Odds"
Note that W is a pure value judgment, compltely idepndent of any consideration of the probability of stae S ot T

The more regrettable A would be if T happens, the higher the probability of S needs to be to justify choosing A.

CONVERTING FROM CRITICAL ODDS TO CRITICAL PROBABILITY
p/ (1-p) > W

P > (1-P)W

P > W - PW

P + PW > W

p > W/(1+W)
Choose Action A if p >  W/ (1+W)

 p/ (1-p) < W

P < (1-P)W

P < W - PW

P + PW < W

p < W/(1+W)
Choose Action B if p < W/ (1+W),

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