Does Game theory consider opportunity cost?

Question

Morton Davis in his book Game Theory: A non-Technical introduction brings the following example of a non-zero sum game (pg.73) :

A Buisness Partnership

A builder and an architect are jointly offered a contract to design and construct a building. They are offered a single lump sum as a payment and are given a week to decide whether they accept. The approval of both is required, so they must decide how the lump sum will be divided. The architect writes to the builder suggesting that the profits be divided equally. The builder replies, committing himself to the contract in writing provided he receives 60 percent of the profits. He informs the architect moreover, that he is going on a two week vacation, that he cannot be reached, and it is up to the architect to accept the contract on these terms or reject it entirely.

The architect feels like he is being exploited. He feels that his services are valuable as the builder's and that each should receive half the profits. On the other hand, the probable profit is large, and under different circumstances, he would consider 40 percent of such a total an acceptable fee. Should he accept the builder's offer? Note that the builder has no more options; on the architect has the choice- to accept or reject the contact. This is reflected in the payoff matrix shown:

$$\left[\begin{matrix}(0.40,0.60)\\(0, 0)\end{matrix} \right]$$

My question :

The way the case seems to be presented is that the architect is forced to choose 40 percent in light of the alternative where there is no profits to be gained. What about the architect's opportunity cost? Isn't there what to be gained from the architect when refusing this split of profits in a 40-60 split as far as opportunity cost in concerned?

Answer 1

Recall that Oppurtunity Cost is referred to the benefit forgone by choosing the other option.

In your example of a single player game, opportunity cost exists since the architect is completely in control of the benefit he receives. In this case the opportunity cost for signing the contract is 0 (and conversely -40 for not signing the contract).

However this is only applicable in a single person game.

In a case where the business owner and Architect were actively negotiating, it would appear that there would not be any practical ramifications for considering opportunity cost because the payoffs change depending on the other player and can only be seen after the fact.

for example consider a single period 2x2 game such that: $$\left[\begin{matrix}(10,5)&(2,2)\\(2,2)&(5,10)\end{matrix} \right]$$

From the horizontal/vertical player's position his payoffs range from 10,5 and 2. He does not choose with respect to costs, he chooses based on what he thinks the payoffs will be.

---

To answer the broader question which you are probably wondering about.
Opportunity cost is only considered when a buy in for a specific game is necessary

You'd have to consider expected value of a game compared to the cost. (however that only provides a fair price when the game is played infinity, you would also have to consider how many times you'd have to play a game and end up with a positive profit) using this formula:

$$n=\left(\frac{1.96\sigma}{p-\mu}\right)^2$$

I suggest reading up on Martingales and the St. Petersberg Paradox for more info.

Answer 2

In the case of this problem, you are assuming that the opportunity cost of the architect is zero (and the same for the builder), since you are saying that his payoff from refusing the contract is zero.

Remember that this is a theoretical game. You could include a different payoff for the opportunity cost, but it would be a different problem from the one you are presenting.