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