Examine two separate cases for payoff matrices

[davidrpugh]

As noted by @markeschaffer in issue #13, we should look at two separate cases for payoff matrices. In the first case, the IPD condition (2 * Pi_AA > Pi_aA + Pi_Aa) is satisfied and in the second case it is not. Primary research question is whether or not altruism can be supported as an equilibrium in the 1M2F model when the IPD condition is satisfied.

I am running simulations at the moment with a payoff matrix of Pi_aA = 5 > Pi_AA = 4 > Pi_aa = 2 > Pi_Aa=1 with a slightly denser grid of points (~73,000 simulations in total). Note that the IPD condition is satisfied for this payoff matrix. We will see what happens.

It would probably be a good idea to settle on our two payoff matrices sooner rather than later. Aren't there "standard" payoff matrices that are frequently used?

[markeschaffer]

5 > 4 > 2 > 1 is going to run into the problem that Paul spotted:

5-4 = 2-1

and "the relative payoff to selfishness versus altruism is independent of the number of selfish people in the population, so no returns to scarcity".

So in addition to the IPD condition

2*Pi_AA > (Pi_aA + Pi_Aa)

which in this model tells us about the returns to hierarchy vs. egalitarianism, we also need to pay attention to

(Pi_aA - Pi_AA) > (Pi_aa - Pi_Aa)

The term on the left is the difference in payoffs to choosing a vs A when you are facing an A. The term on the right is the difference in payoffs to choosing a vs A when you are facing an a.

Plus we have the condition we discussed yesterday, which is that if we require

Pi_Aa >= 1

which guarantees that there are always enough females in the adoption pool for males to get mates.

[davidrpugh]

Damn it. @markeschaffer well spotted! I will try again

payoffs = {'PiaA': 6.0, 'PiAA': 4.0, 'Piaa': 2.0, 'PiAa': 1.0}

We should get returns to scarcity as 6 - 4 = 2 > 2 - 1 = 1; IPD condition is satisfied as 2 * 4 =8 > 6 + 1 = 7; and we have guaranteed that there are always enough females in the adoption pool for males to get mates. Does this mean we have ruled out the depletion problem? I thought from our discussion yesterday we could not solve the depletion problem simply by restricting the payoff matrix.

Note that in order to destroy the IPD condition we need only make the payoff PiaA > 7. Alternatively, we could lower the payoff PiAA = 3.

[markeschaffer]

On your last point – no, it doesn’t rule out the depletion problem. A minimal condition is that there are enough females in aggregate for all males to find mates. The depletion problem is specific to matching and adoption pools, and arises when there aren’t enough females in the pool for males to get mates.

[davidrpugh]

We now have two separate branches for parameter sweeps: one branch called ipd-condition-not-satisfied for parameter sweeps with payoff matrices that do not satisfy the IPD condition; and ipd-condition-satisfied for parameter sweeps with payoff matrices that do satisfy the IPD condition. All of the plots should be dynamically linked so that they will update whenever I push to either branch.