How far can we go with rationalization?

1) ISD, strictly dominated strategies were predictable as they were erased one by one

2) cross out one for rationalizability, never best response

2-player games make the same predictions as ISD and rationalizability

The concept of Nash equilibrium emerged for people’s beliefs.

\(BR_{1}(\theta_{1})\): 1’s belief about 2’s behavior

\(\theta_{1}\ = s_{2}^{*}\): correct belief

\(BR_{1}(S_{2}^{*}) = S_{1}^{*}\)

\(BR_{2}(\theta_{2}) = S_{2}^{*}, \theta_{2} = s_{1}^{*}\)

2. Normal-form games

2.5 Application 1: Oligpoly, Tariffs, and Crime (Chapter 10)

### Cournot Duopoly

EXAMPLE 2.17. Two firms compete by choosing quantity produced in a market, where the demand function is given by \(P(q_{1}, q_{2}) = 12 – q_{1} – q_{2}\). The two firms have an identical cost function \(C(q_{i}) = 4q_{i}\). Find the Nash equilibrium of this game.

Firm 1’s profit function,

\(u_{1}(q_{1}, q_{2}) = (12 – q_{1} – q_{2})q_{1} – 4q_{1}\)

From the FOC, we obtain **\(BR_{1}(q_{2})\)** = \(\frac{8-q_{2}}{2}\)

By symmetry,

**\(BR_{2}(q_{1})\)** = \(\frac{8-q_{1}}{2}\)

N = {1, 2}

\(S_{1}\) = [0, ∞) = \(S_{2}\)

\(\Pi_{1}\) = Rev – cost = p∙q – 4∙\(q_{1}\) = \((12 – q_{1} – q_{2})q_{1} – 4q_{1}\)

\(\frac{\partial \Pi_{1}}{\partial q_{1}} = 12 – 2q_{1} – q_{2} – 4 = 0\) ⇔ \(q_{1} = \frac{8-q_{2}}{2}\)

Slutsky’s equation

```
f(q) = (8-q)/2, f(q) = 8-2q, q=(0,8)
x-axis: q1, y-axis: q2
```

- Reference: Chang-Koo Chi, (9/50) Game Theory and Applications 3 – Application I, Jul 2, 2020, https://youtu.be/1f-mYmkM444