How should we expect Bayesian rational players to behave in a normal-form game?

We will consider a sequence of increasingly restrictive methods for analyzing normal form games.

Iterated Strict Dominance | Rationality |

Rationalizability | Rationality + Reasonable beliefs |

Nash Equilibrium | Rationality + Correct beliefs |

### Strictly dominant strategies

DEFINITION 2.1. A pure strategy \(s_{i}\) of player i is **strictly dominant** if

\(u_{i}(s_{i}, \sigma_{j})\) > \(u_{i}(s_{i}^{\prime}, \sigma_{j})\) for all \(s_{i}^{\prime} \neq s_{i}\) and \(\sigma_{j} \in \Delta (S_{2})\).

When, for any strategy \(s_{i}\) available to player i, strategy \(s_{i}\) gives player i the highest expected payoff, regardless of what opponent player j does.

This means that, \(s_{i}\) is strictly dominant → \(s_{i}\) has the highest payoff → \(s_{i}\) is the best strategy

That is, a strictly dominant strategy yields the highest payoff to player i no matter what his opponent plays

EXAMPLE 2.5.

1 \ 2 | L | R |

U | 2, 3 | 5, 0 |

D | 1, 0 | 4, 3 |

Strategy U yields a strictly higher payoff to P1 than D, regardless of P2’s behavior

For each L and R, strategy U is better than D (∵ 2 > 1, 5 > 4)

THEOREM 2.2. Pure strategy \(s_{i}\) is strictly dominant for player i if and only if

\(u_{i}(s_{i}, \theta_{i})\) > \(u_{i}(s_{i}^{\prime}, \theta_{i})\) for all \(s_{i}^{\prime} \neq s_{i}\) and for all \(\theta_{i} \in \Delta S_{j}\).

A rational player would play a strictly dominant strategy whenever it exists

EXAMPLE 2.6.

1 \ 2 | C | D |

C | 2, 2 | 0, 3 |

D | 3, 0 | 1, 1 |

Joint payoffs are maximized if both players **Cooperate**. But regardless of what player 2 does, player 1 is better off by **Defecting**. The same is true for player 2. Hence D becomes a rational choice for both players.

Players 1 and 2 both think that C, C is better than D, D, but they know that D is the better strategy, so the C, C strategy is not played.

### Efficiency

A strategy profile s = \((s_{1}, s_{2})\) is called (Pareto) **efficient** if there is no other

s’ = \((s_{1}^{\prime}, s_{2}^{\prime})\) such that

\(u_{i}(s^{\prime})\) ≥ \(u_{i}(s)\) for every player i

and the inequality is strict for at least one player

If you can increase your own payoff without harming others, the previous ones are inefficient.

strategy profile is a collection of strategies that each player can do.

EXAMPLE 2.7.

1 \ 2 | L | R |

U | 4, 1 | 0, 2 |

M | 0, 0 | 4, 0 |

D | 1, 3 | 1, 2 |

(D, R) is not efficient because (D, L) Pareto improves the players’ payoffs

This is because there are other strategy profiles that can be correct improved

The efficient strategy profiles in this game are (U, L), (D, L)

EXAMPLE 2.8.

1 \ 2 | L | C | R |

U | 8, 3 | 0, 4 | 4, 4 |

M | 4, 2 | 1, 5 | 5, 3 |

D | 3, 7 | 0, 1 | 2, 0 |

There is no strictly dominant strategy for both players. How can we make a prediction in this case?

There is no best strategy for players 1 and 2. In other words, there is no strictly dominant strategy for both players 1 and 2.

If player 2 chooses L, U is better, but M is better with C or R

If player 1 chooses D, L is better, but C is better with U or M

### Strictly dominated strategies

DEFINITION 2.3. A pure strategy \(s_{i}\) of player i is **strictly dominated** (by \(\sigma_{i}\)) if there exists a strategy \(\sigma_{i}\) satisfying

\(u_{i}(\sigma_{i}, s_{j})\) > \(u_{i}(s_{i}, s_{j})\) for all \(s_{j}\).

In words, if a strategy \(s_{i}\) is strictly dominated, then player i has an alternative that is strictly better than \(s_{i}\) no matter what player j does.

The next theorem is an analogue of Theorem **2.2**:

THEOREM 2.4. A strategy \(s_{i}\) is strictly dominated if and only if there exists a strategy \(\sigma_{i}\) such that

\(u_{i}(\sigma_{i}, \theta_{i})\) > \(u_{i}(s_{i}, \theta_{i})\) for all player i’s beliefs \(\theta_{i}\).

Hence a rational player never chooses strictly dominated strategies.

A strategy that is not dominated by any pure strategy may be dominated by a **mixed strategy**

1 \ 2 | L | R |

U | 4, 1 | 0, 2 |

M | 0, 0 | 4, 0 |

D | 1, 3 | 1, 2 |

If a pure strategy \(s_{i}\) is strictly dominated, then any mixed strategies \(\sigma_{i}\) with \(\sigma_{i}(s_{i})\) > 0 are also dominated

\(\frac{1}{3}M + \frac{2}{3}D\) is strictly dominated in this example

D is strictly dominated by \(\frac{1}{2}U + \frac{1}{2}M\)

\(\frac{1}{3}M + \frac{2}{3}D\) < \(\frac{1}{3}M + \frac{2}{3}(\frac{1}{2}U + \frac{1}{2}M) = \frac{2}{6}U + \frac{2}{3}M\)

### Iterated Strict Dominance

Theorem **2.4** tells us that P2 would never choose strategy X in the following game:

1 \ 2 | X | Y | Z |

A | 3, 3 | 0, 5 | 0, 4 |

B | 0, 0 | 3, 1 | 1, 2 |

Since **rationality is common knowledge**, P1 knows P2 wouldn’t play X.

Also, P2 knows P1 knows this fact, i.e.,

P2 knows P1 knows P2 wouldn’t play X …

Hence strategy X is eliminated by *both* players

Interactive knowledge can be used to make a sharp prediction in this way

After eliminating X, A is then strictly dominated by B

P2 knows this, and P1 knows P2 knows this …

Eliminating A, Y is then strictly dominated by Z

P1 knows this, and P2 knows P1 knows this …

We end up with a unique strategy profile (B, Z)

▶︎ This procedure is called **iterated strict dominance** (ISD)

▶︎ If the procedure leaves only one strategy profile, the game is *solvable* by ISD

▶︎ ISD only hinges upon common knowledge of rationality and in particular, it is independent of players’ beliefs

### Examples of ISD

To see which strategies survive the process, it is enough

(1) First iteratively remove all dominated pure strategies

(2) When no further pure strategies can be removed, check all remaining mixed strategies

EXAMPLE 2.10.

1 \ 2 | L | C | R |

U | 8, 3 | 0, 4 | 4, 4 |

M | 4, 2 | 1, 5 | 5, 3 |

D | 3, 7 | 0, 1 | 2, 0 |

To be strictly dominated, it must be a strict inequality. For U, it is ambiguous that C and R are equal, so it is not strictly dominated.

strategy D is strictly dominated by M (relative to player 1)

strategy L is strictly dominated by C (relative to player 2)

strategy U is strictly dominated by M (relative to player 1)

R is removed from C and R by player 2

Therefore, (M, C) is a prediction that can be made by ISD.

1 \ 2 | X | Y | Z |

U | 5, 1 | 0, 4 | 1, 0 |

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

D | 3, 3 | 4, 4 | 2, 5 |

by player 2, the strategy X is strictly dominated by a half mix of Y and Z. (using mixed strategy)

by player 1, the strategy U is strictly dominated by D

by player 2, the strategy Y is strictly dominated by Z

Therefore, the prediction by ISD is (M, Z).

THEOREM 2.5. The set of strategies that remains after iteratively removing strictly dominated strategies does **not depend on the elimination order**.

EXAMPLE 2.11 (EXERCISE 6.1, p63).

1 \ 2 | W | X | Y | Z |

U | 3, 6 | 4, 10 | 5, 0 | 0, 8 |

M | 2, 6 | 3, 3 | 4, 10 | 1, 1 |

D | 1, 5 | 2, 9 | 3, 0 | 4, 6 |

by player 2, the strategy Z is strictly dominated by X

by player 1, the strategy M, D is strictly dominated by U

by player 2, the strategy W, Y is strictly dominated by X (X is strictly dominant)

Therefore, the prediction by ISD is (U, X).

Alternatively, from player 1’s perspective, U * 0.6 + D * 0.4 is always greater than M, so D is deleted

### Weak Dominance

DEFINITION 2.6. A strategy \(\sigma_{i}\) is **weakly dominated** if there exists another strategy \(\sigma_{i}^{\prime}\) such that for all \(s_{j}\),

\(u_{i}(\sigma_{i}^{\prime}, s_{j})\) ≥ \(u_{i}(\sigma_{i}, s_{j})\) with strict inequality for some \(s_{j}\).

A strategy \(s_{i}\) is **weakly dominant** if \(s_{i}\) weakly dominates all other strategies.

In contrast to strict dominance, weakly dominated strategies *cannot* be eliminated by common knowledge of rationality alone

1 \ 2 | L | M |

X | 3, 5 | 3, 5 |

Y | 7, 0 | 1, 1 |

L is weakly dominated by M, and M becomes a weakly dominant strategy.

But if player 2 believes that player 1 will definitely choose X, player 2 can play strategy L (since both L and M are good strategies)

In this case, it becomes dependent on belief, but strict dominated is independent of belief.

Player 2 may reasonably select L if she believes that player 1 will choose X for sure

Strategy L can be eliminated by additional conditions,

e.g., the principle of **cautiousness**

In contrast to iterated strict dominance, we may have *different* predictions depending on the elimination order

1 \ 2 | L | R |

T | 5, 1 | 4, 0 |

M | 6, 0 | 3, 1 |

B | 6, 4 | 4, 4 |

by player 1, the strategy M, T is weakly dominated by B

for player 2, there is no dominant strategy because (B, L) and (B, R) are the same.

by player 1, the strategy M is weakly dominated by B

by player 2, the strategy R is weakly dominated by L

by player 1, the strategy T is weakly dominated by B

Therefore, the prediction by ISD is (B, L).

by player 1, the strategy T is weakly dominated by B

by player 2, the strategy L is weakly dominated by R

by player 1, the strategy M is weakly dominated by B

Therefore, the prediction by ISD is (B, R).

- Reference: Chang-Koo Chi, (5/50) Game Theory and Applications 2 – Strict/weak dominance, Jul 1, 2020, https://youtu.be/EMtTVIfH-qc