Governmentally mandated paternity testing... (Part III)

Starting with:

Payoff matrix with formulas.
	Female payoffs		Male payoffs		Offspring payoffs
	Prosocial	Financial	Prosocial	Financial	Offspring	Financial
Marriage (2)	P1 + P2 + bP1P2	($1+$2)/2	P1 + P2 + bP1P2	($1+$2)/2
Separate (1+1)	P1	$1	P2	$2
Marriage (3)	P1 + P2 + bP1P2	pCON{[($1+$2)/3]} + (1 - pCON){[($1+$2)/2]}	P1 + P2 + bP1P2	pCON{[($1+$2)/3]} + (1 - pCON){[($1+$2)/2]}	pCON{[(P1+P2)/2] + [(G1+G2)/2] + bP1P2G1G2}	pCON[($1+$2)/3]
Alone/Abandoned	P1	pCON($1/2) + (1 - pCON)($1)	P2	$2	pCON{[(P1/2) + [(G1+G2)/2]}	pCON[($1)/2]
Alone/Paying	P1	pCON[$1/(2 - c)] + (1 - pCON)($1)	P2	pCON[$2/(1 + c)] + (1 - pCON)($2)	pCON{[(P1/2) + [(G1+G2)/2]}	pCON{{$1-[$1/(2 - c)]} + {$2-[$2/(1 + c)]}}
Alone/Adopted	P1	$1	P2	$2	pCON{ 0.75 + [(G1+G2)/2] + 0.56b(G1G2)}	Max: $1 or $2
Grayed cells represent non-reproductive outcomes.

Prosocial Payouts


Examining the payouts, the prosocial payouts fall into two categories: the baseline self-values and the two marriage categories allowing access to the benefits of a mate.  Subsituting the definite P₁ & P₂ for relative values P_i & P_j in the formala, we set P_i + P_j + bP_iP_j = P_i in order to identify what values of b and P_j increase the payoff for marriage, we find all positive values of b and P_j increase the payoffs and negative values of b decrease the payoff.

Prosocial payoffs.
	Worse Payoffs	Equivalent Payoffs	Increased Payoffs
Marriage	b < 0	b = 0 and/or Pj = 0	b > 0 and Pj > 0
All Others		Baseline (Pi)
Grayed cells represent inaccessible outcomes.

Financial Payouts


Using a similar approach to the fiscal payouts--selecting the base finances for each individual $₁ & $₂ as $_i & $_j and setting each payout equivalent to $_i--we calculated the following relationships for females:

Female financial payoffs (base).
	Worse Payoffs	Equivalent Payoffs	Increased Payoffs
Marriage (2)	$1 > $2	$1 = $2	$1 < $2
Marriage (3)	$1 > [(3-pCON)/(3+pCON)]$2	$1 = [(3-pCON)/(3+pCON)]$2	$1 < [(3-pCON)/(3+pCON)]$2
Separate		Baseline
Alone/Abandoned	pCON > 0	pCON = 0
Alone/Paid	c < 0	c = 0
Alone/Adopted		Baseline
Grayed cells represent inaccessible outcomes.

One issue--and one that will figure into the analysis--is the proportion of the baby's resources [d] the mother has access to.  In general, any increase in [d] where the baby's resources are not zero increases the woman's payoff and the probability of conception (p_CON).  Specifically, for marriage with child and woman alone with father paying increase the baby's resource with increased father's resources ($₂).  Adoption essentially converts [d] to zero.  For alone/abandoned, however, the maximum resources would equal the woman's income and result in essentially neglect to the baby.  Abortion--as will be discussed--converts the p_CON to 0 at the female's option. 

Female financial payoffs (shared resources).
	Worse Payoffs	Equivalent Payoffs	Increased Payoffs
Marriage (2)	$1 > $2	$1 = $2	$1 < $2, d > 0
Marriage (3)	$1 > [(3-pCON)/(3+pCON)]$2	$1 = [(3-pCON)/(3+pCON)]$2	$1 < [(3-pCON)/(3+pCON)]$2,d > 0
Separate		Baseline
Alone/Abandoned	pCON > 0	pCON = 0	d > 0
Alone/Paid	c < 1	c = 1	d > 0, $2> 0, c > 0
Alone/Adopted		Baseline
Grayed cells represent inaccessible outcomes.

Male payoffs assume the mother is the custodial parent.  There are only four types of payoff: both marriages, alone/paying, and a cluster including all others due to non-shared resources.  The male payoff table looks like this:

Male financial payoffs.
	Worse Payoffs	Equivalent Payoffs	Increased Payoffs
Marriage (2)	$2 > $1	$2 = $1	$2 < $1
Marriage (3)	$2 > [(3-pCON)/(3+pCON)]$1	$2 = [(3-pCON)/(3+pCON)]$1	$2 < [(3-pCON)/(3+pCON)]$1
Alone/Paid	c > 0 and pCON > 0	c = 0 or pCON = 0
All Others		Baseline
Grayed cells represent inaccessible outcomes.

We can now construct a decision matrix for both sexes.