Renters and Homeowners

Instead of doing productive things I've been reading Games and Decisions by Duncan Luce. I like the fact that it covers the arguments against the normative assumptions of game theory, although having to look in the appendix's for the formal statement of the theory is a pain. Still it inspired this simplified model of renters and homeowners as a zero-sum game.

Players:

Renters (R)
Homeowners (H)

Moves:

Develop high end apartments (HIGH)
Develop low end apartments (LOW)
Develop no apartments (NO)

Payout for Renters:

High income apartments built: 1
Low income apartments built: 2
No income apartments built: 0

This assignment of payouts is arbitrary. However I think its broadly justifiable, since anytime apartments are built it should increase the supply of apartments and decrease their cost, and more renters can afford low end apartments than high end apartments.

Lets also assume that the only political players in the city are renters and homeowners. The political power of renters is $r$, the political power of the home owners is $l$, and the total political power in the city is $$ c=r+l $$

Now assume that the actual payout achieved by the renters is a function of the political power put behind a given move, so if both renters and homeowners decide to build high end apartments the renters payout is $1*c$. This assumption produces the following payout matrix for renters,

HIGH

LOW

HIGH

$1c$

$1r+2h$

$1r+0h$

LOW

$2r+1h$

$2c$

$2r+0h$

$0r+1h$

$0r+2h$

$0c$

which simplifies to,

HIGH

LOW

HIGH

$c$

$r+2h$

$r$

LOW

$2r+h$

$2c$

$2r$

$h$

$2h$

$0$

Now given $c=l+r$ it follows that $c\geq r$ and $c \geq l$. Therefore the equilibrium point of this zero-sum game is renters advocating to build low end apartments and homeowners advocating for no development.

Takeaways:

At the equilibrium point renters always advocate for low income apartments and attempt to maximize their political power, $r$.
At the equilibrium point homeowners always advocate for no development and attempt to minimize the political power $r$ of renters.

I am surprised how well this simple model captures the real world dynamics between renters and homeowners, even if that surprise ignores a bit of circular reasoning given the model was constructed to model these dynamics.