{"id":1008,"date":"2023-05-28T23:50:23","date_gmt":"2023-05-28T14:50:23","guid":{"rendered":"https:\/\/saraheee.com\/?p=1008"},"modified":"2023-05-29T01:25:33","modified_gmt":"2023-05-28T16:25:33","slug":"game-theory-2-chap05-strict-weak-dominance","status":"publish","type":"post","link":"https:\/\/saraheee.com\/ko\/2023\/05\/game-theory-2-chap05-strict-weak-dominance\/","title":{"rendered":"Game Theory 2 \u2013 chap05. Strict\/weak dominance"},"content":{"rendered":"

How should we expect Bayesian rational players to behave in a normal-form game?<\/p>\n\n\n\n

We will consider a sequence of increasingly restrictive methods for analyzing normal form games.<\/p>\n\n\n\n

Iterated Strict Dominance<\/td>Rationality<\/td><\/tr>
Rationalizability<\/td>Rationality + Reasonable beliefs<\/td><\/tr>
Nash Equilibrium<\/td>Rationality + Correct beliefs<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Strictly dominant strategies<\/h3>\n\n\n\n

DEFINITION 2.1<\/mark>. A pure strategy \\(s_{i}\\) of player i is strictly dominant<\/mark><\/strong> if<\/p>\n\n\n\n

\\(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})\\).<\/p>\n\n\n\n

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.<\/mark><\/p>\n\n\n\n

This means that, \\(s_{i}\\) is strictly dominant \u2192 \\(s_{i}\\) has the highest payoff \u2192 \\(s_{i}\\) is the best strategy<\/mark><\/p>\n\n\n\n

That is, a strictly dominant strategy yields the highest payoff to player i no matter what his opponent plays<\/p>\n\n\n\n

EXAMPLE 2.5<\/mark>.<\/p>\n\n\n\n

1 \\ 2<\/td>L<\/td>R<\/td><\/tr>
U<\/td>2, 3<\/td>5, 0<\/td><\/tr>
D<\/td>1, 0<\/td>4, 3<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Strategy U yields a strictly higher payoff to P1 than D, regardless of P2’s behavior<\/p>\n\n\n\n

For each L and R, strategy U is better than D (\u2235 2 > 1, 5 > 4)<\/mark><\/p>\n\n\n\n

THEOREM 2.2<\/mark>. Pure strategy \\(s_{i}\\) is strictly dominant for player i if and only if<\/p>\n\n\n\n

\\(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}\\).<\/p>\n\n\n\n

A rational player would play a strictly dominant strategy whenever it exists<\/p>\n\n\n\n

EXAMPLE 2.6<\/mark>. <\/p>\n\n\n\n

1 \\ 2<\/td>C<\/td>D<\/td><\/tr>
C<\/td>2, 2<\/td>0, 3<\/td><\/tr>
D<\/td>3, 0<\/td>1, 1<\/td><\/tr><\/tbody><\/table>
Prisoners’ Dilemma<\/figcaption><\/figure>\n\n\n\n

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

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.<\/mark><\/p>\n\n\n\n

Efficiency<\/h3>\n\n\n\n

A strategy profile s = \\((s_{1}, s_{2})\\) is called (Pareto) efficient<\/mark><\/strong> if there is no other
s’ = \\((s_{1}^{\\prime}, s_{2}^{\\prime})\\) such that<\/p>\n\n\n\n

\\(u_{i}(s^{\\prime})\\) \u2265 \\(u_{i}(s)\\) for every player i<\/p>\n\n\n\n

and the inequality is strict for at least one player<\/p>\n\n\n\n

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.<\/mark><\/p>\n\n\n\n

EXAMPLE 2.7<\/mark>.<\/p>\n\n\n\n

1 \\ 2<\/td>L<\/td>R<\/td><\/tr>
U<\/td>4, 1<\/td>0, 2<\/td><\/tr>
M<\/td>0, 0<\/td>4, 0<\/td><\/tr>
D<\/td>1, 3<\/td>1, 2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

(D, R) is not efficient because (D, L) Pareto improves the players’ payoffs<\/p>\n\n\n\n

This is because there are other strategy profiles that can be correct improved<\/mark><\/p>\n\n\n\n

The efficient strategy profiles in this game are (U, L), (D, L)<\/mark><\/p>\n\n\n\n

EXAMPLE 2.8<\/mark>.<\/p>\n\n\n\n

1 \\ 2<\/td>L<\/td>C<\/td>R<\/td><\/tr>
U<\/td>8, 3<\/td>0, 4<\/td>4, 4<\/td><\/tr>
M<\/td>4, 2<\/td>1, 5<\/td>5, 3<\/td><\/tr>
D<\/td>3, 7<\/td>0, 1<\/td>2, 0<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

There is no strictly dominant strategy for both players. How can we make a prediction in this case?<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

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<\/mark><\/p>\n\n\n\n

Strictly dominated strategies<\/h3>\n\n\n\n

DEFINITION 2.3<\/mark>. A pure strategy \\(s_{i}\\) of player i is strictly dominated<\/mark><\/strong> (by \\(\\sigma_{i}\\)) if there exists a strategy \\(\\sigma_{i}\\) satisfying<\/p>\n\n\n\n

\\(u_{i}(\\sigma_{i}, s_{j})\\) > \\(u_{i}(s_{i}, s_{j})\\) for all \\(s_{j}\\).<\/p>\n\n\n\n

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.<\/p>\n\n\n\n

The next theorem is an analogue of Theorem 2.2<\/mark><\/strong>:<\/p>\n\n\n\n

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

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

Hence a rational player never chooses strictly dominated strategies.<\/p>\n\n\n\n

A strategy that is not dominated by any pure strategy may be dominated by a mixed strategy<\/mark><\/strong><\/p>\n\n\n\n

1 \\ 2<\/td>L<\/td>R<\/td><\/tr>
U<\/td>4, 1<\/td>0, 2<\/td><\/tr>
M<\/td>0, 0<\/td>4, 0<\/td><\/tr>
D<\/td>1, 3<\/td>1, 2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

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

\\(\\frac{1}{3}M + \\frac{2}{3}D\\) is strictly dominated in this example<\/p>\n\n\n\n

D is strictly dominated by \\(\\frac{1}{2}U + \\frac{1}{2}M\\)<\/p>\n\n\n\n

\\(\\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\\)<\/p>\n\n\n\n

Iterated Strict Dominance<\/h3>\n\n\n\n

Theorem 2.4<\/mark><\/strong> tells us that P2 would never choose strategy X in the following game:<\/p>\n\n\n\n

1 \\ 2<\/td>X<\/td>Y<\/td>Z<\/td><\/tr>
A<\/td>3, 3<\/td>0, 5<\/td>0, 4<\/td><\/tr>
B<\/td>0, 0<\/td>3, 1<\/td>1, 2<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

Since rationality is common knowledge<\/mark><\/strong>, P1 knows P2 wouldn’t play X.
Also, P2 knows P1 knows this fact, i.e.,<\/p>\n\n\n\n

P2 knows P1 knows P2 wouldn’t play X …<\/p>\n\n\n\n

Hence strategy X is eliminated by both<\/em> players
Interactive knowledge can be used to make a sharp prediction in this way<\/p>\n\n\n\n

After eliminating X, A is then strictly dominated by B
P2 knows this, and P1 knows P2 knows this …<\/p>\n\n\n\n

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)<\/p>\n\n\n\n

\u25b6\ufe0e This procedure is called iterated strict dominance<\/mark><\/strong> (ISD)
\u25b6\ufe0e If the procedure leaves only one strategy profile, the game is solvable<\/em> by ISD
\u25b6\ufe0e ISD only hinges upon common knowledge of rationality and in particular, it is independent of players’ beliefs<\/p>\n\n\n\n

Examples of ISD<\/h3>\n\n\n\n

To see which strategies survive the process, it is enough<\/p>\n\n\n\n

(1) First iteratively remove all dominated pure strategies
(2) When no further pure strategies can be removed, check all remaining mixed strategies<\/p>\n\n\n\n

EXAMPLE 2.10<\/mark>.<\/p>\n\n\n\n

1 \\ 2<\/td>L<\/td>C<\/td>R<\/td><\/tr>
U<\/td>8, 3<\/td>0, 4<\/td>4, 4<\/td><\/tr>
M<\/td>4, 2<\/td>1, 5<\/td>5, 3<\/td><\/tr>
D<\/td>3, 7<\/td>0, 1<\/td>2, 0<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

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.<\/mark><\/p>\n\n\n\n

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<\/mark><\/p>\n\n\n\n

Therefore, (M, C) is a prediction that can be made by ISD.<\/mark><\/p>\n\n\n\n

1 \\ 2<\/td>X<\/td>Y<\/td>Z<\/td><\/tr>
U<\/td>5, 1<\/td>0, 4<\/td>1, 0<\/td><\/tr>
M<\/td>3, 1<\/td>0, 0<\/td>3, 5<\/td><\/tr>
D<\/td>3, 3<\/td>4, 4<\/td>2, 5<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

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<\/mark><\/p>\n\n\n\n

Therefore, the prediction by ISD is (M, Z).<\/mark><\/p>\n\n\n\n

THEOREM 2.5<\/mark>. The set of strategies that remains after iteratively removing strictly dominated strategies does not depend on the elimination order<\/mark><\/strong>.<\/p>\n\n\n\n

EXAMPLE 2.11<\/mark> (EXERCISE 6.1, p63).<\/p>\n\n\n\n

1 \\ 2<\/td>W<\/td>X<\/td>Y<\/td>Z<\/td><\/tr>
U<\/td>3, 6<\/td>4, 10<\/td>5, 0<\/td>0, 8<\/td><\/tr>
M<\/td>2, 6<\/td>3, 3<\/td>4, 10<\/td>1, 1<\/td><\/tr>
D<\/td>1, 5<\/td>2, 9<\/td>3, 0<\/td>4, 6<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

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)<\/mark><\/p>\n\n\n\n

Therefore, the prediction by ISD is (U, X).<\/mark><\/p>\n\n\n\n

Alternatively, from player 1’s perspective, U * 0.6 + D * 0.4 is always greater than M, so D is deleted<\/mark><\/p>\n\n\n\n

Weak Dominance<\/h3>\n\n\n\n

DEFINITION 2.6<\/mark>. A strategy \\(\\sigma_{i}\\) is weakly dominated<\/mark><\/strong> if there exists another strategy \\(\\sigma_{i}^{\\prime}\\) such that for all \\(s_{j}\\),<\/p>\n\n\n\n

\\(u_{i}(\\sigma_{i}^{\\prime}, s_{j})\\) \u2265 \\(u_{i}(\\sigma_{i}, s_{j})\\) with strict inequality for some \\(s_{j}\\).<\/p>\n\n\n\n

A strategy \\(s_{i}\\) is weakly dominant<\/mark><\/strong> if \\(s_{i}\\) weakly dominates all other strategies.<\/p>\n\n\n\n

In contrast to strict dominance, weakly dominated strategies cannot<\/em> be eliminated by common knowledge of rationality alone<\/p>\n\n\n\n

1 \\ 2<\/td>L<\/td>M<\/td><\/tr>
X<\/td>3, 5<\/td>3, 5<\/td><\/tr>
Y<\/td>7, 0<\/td>1, 1<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

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.<\/mark><\/p>\n\n\n\n

Player 2 may reasonably select L if she believes that player 1 will choose X for sure<\/p>\n\n\n\n

Strategy L can be eliminated by additional conditions,
e.g., the principle of cautiousness<\/mark><\/strong><\/p>\n\n\n\n

In contrast to iterated strict dominance, we may have different<\/em> predictions depending on the elimination order<\/p>\n\n\n\n

1 \\ 2<\/td>L<\/td>R<\/td><\/tr>
T<\/td>5, 1<\/td>4, 0<\/td><\/tr>
M<\/td>6, 0<\/td>3, 1<\/td><\/tr>
B<\/td>6, 4<\/td>4, 4<\/td><\/tr><\/tbody><\/table><\/figure>\n\n\n\n

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.<\/mark><\/p>\n\n\n\n

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<\/mark><\/p>\n\n\n\n

Therefore, the prediction by ISD is (B, L).<\/mark><\/p>\n\n\n\n

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<\/mark><\/p>\n\n\n\n

Therefore, the prediction by ISD is (B, R).<\/mark><\/p>\n\n\n\n

<\/p>\n\n\n\n