Governmentally mandated paternity testing... (Part II)

Deriving a payoff matrix
Before we can derive a payoff matrix, we can consider the fact not all combinations of choices are compatible.  For example, all results from a non-reproductive partner cannot produce a baby and therefore rule out all but two of the parental responsibility options.  If we go through the current process model and identify those sets of choices resulting in no baby (as seen marked in gray lines in the diagram below), a pattern can be discerned where non-reproductive relationships can only produce one of two basic outcomes: marriage [MRY] or abandonment in combination with alone [ABA x ALO] as the other outcomes require a baby. 

Additionally, the set of conception options include differences by sex in each option's effects.  The distinction between overt birth control [CDM] and concealed birth control [HBC] really depend on the sex and type of technique applied.  For example, male sterilization is ineffective less than 1% of the time, while typical hormonal birth control usage can result in conceptions during the first year about 8% of the time. 

A second reason for the distinction is the ability to intentionally deceive a partner about one's birth control usage.  This manipulation by one sex or the other primarily effects the usage of other forms of contraception or choice of sexual activity.  By including a basic probability for conception risk and keeping in mind the use of deception, we can estimate the effects of the different birth control options into the payoffs in a more streamlined manner.

Resolution of Male/Female actions regarding conception. Male actions across the horizontal axis, female on the vertical.
	NON	CDM	HBC
NON/Deceptive HBC	pBASE	pCDM	*
CDM	pCDM	*	*
HBC	pHBC	*	*
Shaded cells assumed to be equivalent to a non-reproductive relationship.  Cells marked * have a low enough combined probability of conception as to be treated as nil.  Numbers indicate probability [p] of conception for each combination. Male HBC assumes a successful sterilization.  CDM x CDM assumes a combination of barrier methods.
The following is assumed: pBASE > pCDM > pHBC

Likewise, by including the traits making a difference in partner selection--prosociality, reproductive capability, good genes--mathematically into the outcome analysis, we can simplify the mate selection options into two choices: non-reproductive relationships and potentially reproductive relationships.  Because of the pattern noted in the combinations of options, we can directly connect non-reproductive partner choices to the pair of non-reproductive parental responsibility options.

As in the conception options, there are systematic differences in the combination of parental responsibility options by sex.  In the original concept, males could employ 3 different options and females 3 with only marriage overlapping.  But, consider the combinations as in the table below.

Resolution of Male/Female actions regarding parental responsibility. Male actions across the horizontal axis, female on the vertical.
	ABA	PAY	MRY
ALO	(2/1/-)	(2*/1*/-)
ADO	(1/1/1**)
MRY			(3/-/-)
Shaded cells not considered. Numbers indicate population [#] in output sets, female set first, male--if different--second and infant's--if different--third.
For PAY x ALO, marked with an *, the baby and mother's [$] is adjusted upward by half of the male's [$] and his is reduced by half. For ABA x ADO, marked **, the infant's [$] is assumed to produce a value of 1.00 at cost to neither parent.
The non-reproductive outcomes (ABA x ALO and MRY x MRY) are (1/1/-) and (2/-/-) respectively.

By discarding the possibility of compulsory marriage and acknowledging the fact adoption legally disconnects financial responsibility on the father, the potential 9 combinations are reduced to the 4 above with the 2 non-reproductive options re-added to make 6 total possible outcomes.

This produces our revised process model with options:

Now we can estimate the way to evaluate each pathway.

Prosociality [P_OUT]

Prosociality is assumed to benefit a person through access to social support.  (For this model, financial resources are not included as those are evaluated separately).  For our model, we will consider prosociality as an estimator of "market access" for social support.  For a partner, two things are considered: their basic "market access" and the effect of synergy on combining both partner's social access.

For example, imagine a cocktail party.  Someone with particular social access may gain access to the party, but may not have access to all the cliques involved like the wives of a particular group.  A second person might not be able to access the party, but once there, may be able to enter into cliques the first person couldn't.  

This synergy will be included in the evaluation as follows:

(1)  P_OUT = P₁ + P₂ + bP₁P₂

Where:
P_OUT = The prosociality index of the relationship (0 -> 2+b)
P₁,P₂ = The prosociality of each partner (0 - 1.0)
b = A scalar of how synergetic people are.
In relationships where only one partner exists, P_OUT = P₁. 

Genetic or offspring value [G_OUT]

Like prosociality, development is an interactive process.  To capture the effects of parental support on the baby, the offspring's value considers both the genetic and social effects of the parents and a multiplicative synergetic term assuming the same synergy ratio as the prosocial elements of the outcome. 

Because the basic elements of the offspring are additive, the basic genetic and prosocial value of the infant are based on an inter-parental mean.  The formula below is used to estimate offspring value in outcomes including offspring:

(2a)  G_OUT = p_CON{[(P₁+P₂)/2] + [(G₁+G₂)/2] + bP₁P₂G₁G₂}

Where:
G_OUT = The offspring value index of the baby (0 -> 2+b)
p_CON = The probability of conception (0 - 1.0)
P₁,P₂ = The prosociality of each partner (0 - 1.0)
G₁,G₂ = The prosociality of each partner (0 - 1.0)
b = A scalar of how synergetic people are.

In relationships where only one partner exists, G_OUT = p_CON{[(P₁/2) + [(G₁+G₂)/2]} where it is assumed no synergetic effects are realized with the absent parent. For adoption, assume an "idealized" family where P₁ and P₂ are replaced with 0.75 resulting in Formula (2b):

(2b)  G_OUT = p_CON{ 0.75 + [(G₁+G₂)/2] + 0.56b(G₁G₂₎}
 
Fiscal value [$]

The fiscal value of the relationship does not assume a fixed value and is based on a relative, proportional contribution. Within marriage (Formula 3a) all parties involved are considered to share a common pool of finances.  For non-reproductive relationships, the p_CON is considered to be zero and the same formulas are used. 

(3a)  $_OUT = p_CON{[($₁+$₂)/3]} + (1 - p_CON){[($₁+$₂)/2]}

Where:
$_OUT = The offspring value index of the baby (0 -> 2+b)
p_CON = The probability of conception (0 - 1.0)
$₁,$₂ = The income of each partner.

For relationships other than marriage, the resources are calculated separately with an additional factor [c] used to estimate the proportion of support for the baby by each parent.  Once again, the same formulas with the p_CON adjusted to zero are employed for non-reproductive relationships. 

The third formula (4c) is the estimated resources allocated to the infant.  Normally, these resources will be accessed by the mother (or custodial parent).  The non-sensical inverse is not considered.

(4a)  $_OUT = p_CON[$₁/(2 - c)] + (1 - p_CON)($₁/1)(4b)  $_OUT = p_CON[$₂/(1 + c)] + (1 - p_CON)($₂/1)(4c)  $_OUT = p_CON{{$₁-[$₁/(2 - c)]} + {$₂-[$₂/(1 + c)]}}

The payoff matrix--for each parental responsibility outcome--now appears like this:



Where:
$_OUT = The offspring value index of the baby (0 -> 2+b)
p_CON = The probability of conception (0 - 1.0)
$₁,$₂ = The income of each partner.
c =  The proportion of child support allocated to the father.

For abandonment/alone, the father's resources are $_OUT = $₂ and mother's $_OUT = p_CON($₁/2) + (1 - p_CON)($₁/1).  For adoption, the parents are treated separately (as if p_CON = 0) and the baby's $_OUT will be considered the higher of either parent's base financial resources for estimation purposes.

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.

Next, we'll simplify this table and begin the actual analysis...