Remember the goalie-striker game from our last article? Like nearly every game in the world of game theory, there are mathematically optimal moves for the given players to take. But unlike the prisoner’s dilemma or the miner analogy from Game Theory Part 2, solving for this game requires some slightly heavier math, so here goes:
Let’s start with the 2nd equation, the one outlining the goalie’s expected points for choosing right. If the goalie dives right, we know that some percentage of the time, he’ll have a point deducted. The rest of the time, he gets 3 points.
Here's what this is showing: if the goalie plays right, his expected points is the probability of the striker playing left (Sigma L multiplied by -1) plus the probability of the striker playing right (1 minus Sigma L multiplied by 3). Where did we get (-1) and 3 from? The points the goalie gets from the striker’s respective moves: if the striker plays left, the goalie gets a point deducted, but if the striker plays right, the goalie receives 3 points. This is the goalies expected points for diving right, and we can easily do the same thing for the goalie diving left:
In this case, we got 0 and -2 points from if the striker plays left or right, respectively. Now that we have these 2 equations, we can combine them and solve for Sigma L:
So what this shows is that if the goalie dives left ⅚ (approximately 83%) of the time, and dives right ⅙ (approximately 17%) of the time, then the striker is indifferent between left and right. In other words, this strategy of diving left 83% of the time and diving right 17% of the time is the goalie’s mixed strategy.
We can do the same calculations from the striker’s point of view:
Combining the 2 and solving for Sigma L:
And so, if the striker shoots left ⅔ (approximately 66%) of the time, and shoots right ⅓ (approximately 33%) of the time, then the goalie is indifferent between diving left or right. Remember, both this and the goalie’s equation is to assure that the expected points will be the same regardless if the other player goes left or right. In conclusion, we now have our mixed strategy Nash equilibrium:
Econ IRL:
Economic policy and culture are 2 areas that one might think are largely unrelated before going through this week’s study, in which Professor Natalie Bau examines the effects of government pension plans on traditional kinship practices. The author first looks at matrilocal customs (ones in which married couples live near the female’s family) and patrilocal customs (ones in which married couples live near the male’s family) in Indonesia, and finds that such practices do indeed decrease with the introduction of a new pension program.
Professor Bau also notes both matrilocality and patrilocality encourage educational investment in their respective genders - matrilocality incentivizes financing towards the female and patrilocality incentivizes financing towards the male - and so delves into the parental investment in children. Turns out that these pension plans have the unintended consequence of reducing human capital investment - less money going towards their kids due to the increased neolocality (basically the neutral position between matrilocality and patrilocality) results in less overall human capital investment.
‘Till next time,
SoBasically