2. Normal-form games

2.3 Best Response and Rationalizability (Chapter 6 & 7)

### Best Response

Question: What is the *tightest* prediction that we can make on the basis of rationality?

Answer: Rational players not only avoid dominated strategies but also avoid strategies that are **never a best response**

Applying this idea iteratively, we obtain the set of *rationalizable* strategies

DEFINITION 2.7. Suppose player i has a belief \(\theta_{i} \in \Delta S_{j}\) about player j’s behavior.

Strategy \(s_{i}\) is a **best response** to belief \(\theta_{i}\) if

\(u_{i}(s_{i}, \theta_{i})\) ≥ \(u_{i}(s_{i}^{\prime}, \theta_{i})\) for every \(s_{i}^{\prime}\ \in S_{i}\).

For any belief \(\theta_{i}\) of player i, we denote by \(BR_{i}(\theta_{i})\) the set of best responses.

Observe that the set of best response is a function of the player’s belief

EXAMPLE 2.12. Suppose player 1 forms a belief \(\theta_{1}\) = (1/3, 1/2, 1/6) on player 2’s behavior in the following game:

1 \ 2 | L | C | R |

U | 2, 6 | 0, 4 | 4, 4 |

M | 3, 3 | 0, 0 | 1, 5 |

D | 1, 1 | 3, 5 | 2, 3 |

Under the belief \(\theta_{1}\), player 1’s expected payoff from each action is

\(u_{1}(U, \theta_{1})\) = \(2 \cdot \frac{1}{3} + 0 \cdot \frac{1}{2} + 4 \cdot \frac{1}{6} = \frac{2}{3} + \frac{2}{3} = \frac{4}{3}\)

\(u_{1}(M, \theta_{1})\) = \(3 \cdot \frac{1}{3} + 0 \cdot \frac{1}{2} + 1 \cdot \frac{1}{6} = 1 \cdot \frac{1}{6} = \frac{7}{6}\)

\(u_{1}(D, \theta_{1})\) = \(1 \cdot \frac{1}{3} + 3 \cdot \frac{1}{2} + 2 \cdot \frac{1}{6} = \frac{1}{3} + \frac{3}{2} + \frac{1}{3} = \frac{13}{6}\)

Hence player 1’s set of best responses is \(BR_{1}(\theta_{1})\) = {D}

Given \(\theta_{2}\) = (1/2, 1/4, 1/4), what is player 2’s best response?

\(u_{2}(L, \theta_{2})\) = \(6 \cdot \frac{1}{2} + 3 \cdot \frac{1}{4} + 1 \cdot \frac{1}{4} = 3 + 1 = 4\)

\(u_{2}(C, \theta_{2})\) = \(4 \cdot \frac{1}{2} + 0 \cdot \frac{1}{4} + 5 \cdot \frac{1}{4} = 2 + \frac{5}{4} = \frac{13}{4}\)

\(u_{2}(R, \theta_{2})\) = \(4 \cdot \frac{1}{2} + 5 \cdot \frac{1}{4} + 3 \cdot \frac{1}{4} = \frac{8+5+3}{4} = \frac{16}{4} = 4\)

### Best Responses and Beliefs

Observe that a player’s set of best responses relies upon the player’s beliefs

Playing a best response is not in itself a strategic act

It is **formation of beliefs** that captures the important strategic component in games

- Success in games often hinges on whether you understand your opponent better than he understands you: you need “outfox” him
- Thus the key question boils down to “What is a
*reasonable*belief for your opponent’s behaviors?” - One possible answer is that you anticipate him to choose neither strictly dominated strategies nor never best responses

### Dominance and Best Response

There is a close relations between dominance and best response. To see this, let

\(B_{i}\) = {\(s_{i} \in S_{i}\) | there exists a belief \(\theta_{i}\) such that \(s_{i} \in BR_{i}(\theta_{i})\)}

\(UD_{i}\) = {\(s_{i} \in S_{i}\) | \(s_{i}\) is not strictly dominated}.

1 \ 2 | L (p) | R (1-p) |

U | 6, 3 | 0, 1 |

M | 2, 1 | 5, 0 |

D | 3, 2 | 3, 1 |

\(u_{1}(U, p)\) = 6p + 0(1-p) = 6p

\(u_{1}(M, p)\) = 2p + 5(1-p) = -3p + 5

\(u_{1}(D, p)\) = 3p + 3(1-p) = 3

```
f(p) = 6p, f(p) = -3p+5, f(p) = 3, p=(0,1)
x-axis: p, y-axis: u1
```

6p = 5 – 3p ⇔ 9p = 5 ⇔ p = 5/9

p > \(\frac{5}{9}\) → \(BR_{1}(p)\) = {U}

p < \(\frac{5}{9}\) → \(BR_{1}(p)\) = {M}

p = \(\frac{5}{9}\) → \(BR_{1}(p)\) = {U, M}

P1’s belief \(\theta_{1}\) can be represented by a scalar variable p ∈ [0, 1]

Strategy D is never a best response to player 1 for all *p*. So **\(B_{1}\) = {U, M}**.

One can easily see that U and M are undominated. What about strategy D?

To examine if there is a mixed strategy strictly better than D, let \(\sigma_{1}\) = (x, 1-x, 0)

and compute P1’s expected payoff from \(\sigma_{1}\) given his belief *p*:

xU + (1-x)M = \(\sigma_{1}\)

\(u_{1}(\sigma_{1}, p)\) = x ∙ \(u_{1}\)(U, p) + (1-x)\(u_{1}\)(M, p)

= x ∙ 6p + (1-x)(5-3p)

= 5(1-x) – 3p + 9xp

When x = 1/3, \(u_{1}(\sigma_{1}, p)\) = 10/3, which is greater than 3, the payoff of \(u_{1}\)(D, p)

so strategy D is strictly dominated by mixed strategies U and M.

Strategies that are not dominated by player 1 \(UD_{1}\) = {U, M} = \(B_{1}\)

### Rationalizability**

Observe that if a strategy is strictly dominated, then it is never a best response:

\(B_{i} \subseteq UD_{i}\) for all i

The converse is not true in general, but it turns out that the converse is also true at least in 2-player games

Therefore,

strictly dominated ⇔ never a best response in 2-player games.

DEFINITION 2.8. **Rationalizable strategies** are those that remain after we iteratively remove all strategy that are never a best response to allowable beliefs.

THEOREM 2.9 (BERNHEIM (1984), PEARCE (1984)). In two-player games, a strategies is rationalizable if and only if it survives ISD.

### Rationalizability in 3-player games

In games with more than two-players,

the set of rationalizable strategies ⊂ the set of strategies that survives ISD.

To see this relation, consider the following example:

W | E | W | E | W | E | |||

T | 9, 9, 5 | 0, 8, 2 | T | 9, 9, 4 | 0, 8, 0 | T | 9, 9, 1 | 0, 8, 2 |

B | 8, 0, 2 | 7, 7, 1 | B | 8, 0, 0 | 7, 7, 4 | B | 8, 0, 2 | 7, 7, 5 |

L | C | R |

P3’s strategy C (the middle matrix) is *not* strictly dominated so survives ISD

On the other hand, C is never a best response to any beliefs \(\theta_{3}\), implying that the strategy is not rationalizable

- Reference: Chang-Koo Chi, (6/50) Game Theory and Applications 2 – Rationalizability, Jul 1, 2020, https://youtu.be/Xv21qzNkuXQ