Equilibrium Dominance: Strict and Weak Dominance in Game Theory

Equilibrium Dominance: Solution Concepts

Introduction to Strict Dominance

• First Solution Concept: Strict dominance aims to eliminate strategies a rational player would never choose.

• Contrast with Other Concepts: Unlike methods that directly search for highest payoff strategies, strict dominance focuses on ruling out obviously bad choices.

• Definition: A strategy is strictly dominated if it consistently yields a strictly lower payoff than another available strategy, regardless of the opponent's actions. It is "always a bad idea."

• Iterated Deletion of Strictly Dominated Strategies (IDSDS):

  • Process: Identify and eliminate strictly dominated strategies for one player, then for another, and continue iteratively until no more strategies can be ruled out for any player.

  • Purpose: "Cleans the game" by removing irrational strategies, leading to a reduced matrix that is easier to analyze.

  • Outcomes:

    • Unique Equilibrium: In some games, IDSDS provides a precise prediction by leaving only one strategy profile (cell).

    • Multiple Strategy Profiles: In other games, IDSDS only eliminates some strategies, leaving several possible outcomes and a less precise prediction.

    • No Bite: In some games, IDSDS may not eliminate any strategies for any player, requiring other solution concepts.

Strictly Dominated Strategies

• Goal: Delete strategies that a rational player would never select.

• Recall (Chapter 1): Player $i$'s strategies satisfy $s<em>i, s'</em>i \in S<em>i$ (for all $s'</em>i \ne s<em>i$), and the strategy profile of player $i$'s rivals is $s</em>{-i} \in S_{-i}$.

• Definition 2.1: Strictly Dominated Strategy: Player $i$ finds that strategy $s<em>i$ strictly dominates strategy $s'</em>i$ (where $s'<em>i \ne s</em>i$) if their payoff satisfies ui(si, s{-i}) > ui(s'i, s{-i}) for every strategy profile $s_{-i}$ of player $i$'s rivals.

  • Intuition: Strategy $s<em>i$ yields a higher payoff than $s'</em>i$ regardless of the rivals' choices. Therefore, $s'_i$ yields a strictly lower payoff and a rational player should never choose it. It can be removed from consideration.

• Tool 2.1: How to find strictly dominated strategies (two-player games):

  1. Focus on one strategy of the column player ($s<em>2$, one specific column). Find a strategy $s'</em>1$ for the row player that yields a strictly lower payoff than some other strategy $s<em>1$ (i.e., u1(s1, s2) > u1(s'1, s_2)).

  2. Repeat step 1 for all other possible strategies of the column player (all other columns), checking if the payoff inequality still holds (u1(s1, s'2) > u1(s'1, s'2)).

  3. If $s'<em>1$ yields a strictly lower payoff than $s</em>1$ for all possible strategies of player 2, then $s'<em>1$ is strictly dominated by $s</em>1$. Otherwise, player 1 does not have a strictly dominated strategy without considering randomizations.

  • Analogous Method for Column Player: Fix the row player's strategy (one specific row) and compare payoffs for the column player (second component) across columns.

Iterated Deletion of Strictly Dominated Strategies (IDSDS)

• Tool 2.2: Applying IDSDS:

  • Step 1 (Rationality): A player deletes their own strictly dominated strategies from their original strategy set ($S<em>i \rightarrow S'</em>i$).

  • Step 2 (Common Knowledge of Rationality): Every player, knowing that opponents are also rational, identifies and deletes strictly dominated strategies for their opponents ($S<em>j \rightarrow S'</em>j$).

  • Step 3: A player re-evaluates their own strategies in light of the opponent's reduced strategy set ($S'<em>j$). Some of the player's previously undominated strategies ($S'</em>i \times S'<em>j$) might now become strictly dominated due to the elimination of rival strategies. This leads to a further reduced set ($S''</em>i$).

  • Step 4 & beyond (Step k): This iterative process continues, eliminating newly dominated strategies for either player, until no more strategies can be deleted. The remaining strategies are the strategy profiles surviving IDSDS.

  • Summary: Rationality helps a player delete their own strictly dominated strategies. Common knowledge of rationality helps them anticipate rivals' deletions, which can make new strategies dominated in subsequent rounds.

• Example 2.1 (Output Game):

  • Matrix 2.1 shows firms choosing High (H), Medium (M), or Low (L) output.
    | Firm 2 | h | m | l |
    |:---|:---:|:---:|:---:|
    | Firm 1 H | 2, 2 | 3, 1 | 5, 0 |
    | M | 1, 3 | 2, 2 | 2.5, 2.5 |
    | L | 0, 5 | 2.5, 2.5 | 3, 3 |

  • For Firm 1, strategy L is strictly dominated by H: (0 < 2 for h), (2.5 < 3 for m), and (3 < 5 for l).

  • For Firm 1, strategy M is also strictly dominated by H: (1 < 2 for h), (2 < 3 for m), and (2.5 < 5 for l). Thus, M and L are both strictly dominated by H. (Due to symmetric payoffs, m and l would be strictly dominated by h for Firm 2).

• Example 2.2 (IDSDS Yields Unique Equilibrium):

  • Initial Matrix 2.2a:
    | Firm 2 | h | l |
    |:---|:---:|:---:|
    | Firm 1 H | 4, 4 | 0, 2 |
    | M | 1, 4 | 2, 0 |
    | L | 0, 2 | 0, 0 |

  • Round 1 (Firm 1): L is strictly dominated by M (e.g., u1(M, h)=1 > u1(L, h)=0 and u1(M, l)=2 > u1(L, l)=0). Row L is deleted.

  • Round 2 (Firm 2): With L deleted, for Firm 2, l is strictly dominated by h (e.g., u2(H, h)=4 > u2(H, l)=2 and u2(M, h)=4 > u2(M, l)=0). Column l is deleted.

  • Round 3 (Firm 1): With l deleted, for Firm 1, M is strictly dominated by H (e.g., u1(H, h)=4 > u1(M, h)=1). Row M is deleted.

  • Result: A unique equilibrium (H, h) with payoffs (4, 4).

Properties of IDSDS Verification

• 2.3.1 Does the Order of Deletion Matter in IDSDS? No.

  • This is a crucial property: deleting strategies in a different order (e.g., starting with player 2 instead of player 1) yields the same set of strategy profiles surviving IDSDS.

  • Example 2.3: Rerunning Example 2.2 starting with Firm 2 confirms the same unique equilibrium (H, h).

• 2.3.2 Deleting More Than One Strategy at a Time:

  • It is permissible to delete all strictly dominated strategies for a player in a single step, rather than one by one, which saves time.

  • If one strategy strictly dominates all other strategies for a player, all those other strategies can be removed simultaneously.

• 2.3.3 Multiple Equilibrium Predictions:

  • Some games are "dominance solvable" where IDSDS yields a unique equilibrium.

  • However, IDSDS often only eliminates some strategies, leaving multiple strategy profiles remaining.

  • Example 2.4: After several rounds, IDSDS yields four surviving profiles: IDSDS = ${(M, h), (M, m), (L, h), (L, m)}$. The order of deletion still doesn't affect the final set of surviving profiles.

Allowing for Randomizations to Bring IDSDS Further

• The "No Bite" Scenario: In games like Battle of the Sexes or Game of Chicken, IDSDS may not eliminate any strategies if players are restricted to "pure strategies" (choosing a strategy with 100% probability).

• Introducing Randomization (Mixed Strategies): If players can choose strategies with probabilities ($q \in (0,1)$), IDSDS can gain "bite."

• Example 2.5:

  • Initial Matrix 2.8: Firm 1 has strategies H, M, L; Firm 2 has h, m, l. No strictly dominated pure strategies exist for either player.

  • Firm 1 Randomizes: Let Firm 1 choose L with probability $q$ and M with probability $(1-q)$. This mixed strategy is denoted $\sigma = qL + (1-q)M$.

  • Expected Utilities: Firm 1's expected payoff from $\sigma$ depends on Firm 2's choice:

    • $EU_1(\sigma|h) = q(10) + (1-q)(4) = 4 + 6q$

    • $EU_1(\sigma|m) = q(6) + (1-q)(0) = 6q$

    • $EU_1(\sigma|l) = q(4) + (1-q)(6) = 6 - 2q$

  • Condition for Dominance: For $\sigma$ to strictly dominate H, its expected payoff must be strictly greater than H's payoff across all of Firm 2's strategies:

    • 4 + 6q > 0 \implies q > -2/3 (always true for $q \in (0,1)$).

    • 6q > 4 \implies q > 2/3.

    • 6 - 2q > 4 \implies q < 1 (always true for $q \in (0,1)$).

  • Result: If Firm 1 chooses any $q \in (2/3, 1)$, the mixed strategy $\sigma$ strictly dominates pure strategy H. H can then be deleted.

  • Further Iteration: After deleting H, new strictly dominated pure strategies appear for Firm 2 (h is strictly dominated by m). This process can continue.

  • Conclusion: Randomization helped reduce the number of equilibrium outcomes from 9 to 4, demonstrating that it can make IDSDS more effective. If IDSDS still has no bite, even with randomization, other concepts like Nash equilibrium are needed.

Evaluating IDSDS as a Solution Concept

• 1. Existence? Yes. At least one strategy profile always survives IDSDS, even if it's the entire game matrix.

• 2. Uniqueness? No. IDSDS frequently yields multiple equilibrium outcomes, failing the strict uniqueness criterion.

• 3. Robust to Small Payoff Perturbations? Yes. Small changes (e.g., $0.001$ or infinitesimally small $\epsilon \rightarrow 0$) to payoffs do not alter strict dominance relationships. If a strategy $s<em>i$ strictly dominates $s'</em>i$, it will continue to do so after a minor perturbation.

• 4. Socially Optimal? No. IDSDS outcomes are not guaranteed to be socially optimal. The Prisoner's Dilemma is a prime example.

Weakly Dominated Strategies

• Definition 2.4: Weakly Dominated Strategies: Player $i$ finds that strategy $s<em>i$ weakly dominates strategy $s'</em>i$ if:

  1. $u<em>i(s</em>i, s<em>{-i}) \ge u</em>i(s'<em>i, s</em>{-i})$ for every strategy profile $s_{-i}$ of player $i$'s rivals.

  2. ui(si, s{-i}) > ui(s'i, s{-i}) for at least one strategy profile $s_{-i}$.

  • Intuition: $s<em>i$ is at least as good as $s'</em>i$ in all situations, and strictly better in at least one situation. Requirement #2 ensures there isn't a complete tie across all outcomes.

• Relationship to Strict Dominance: If a strategy is strictly dominated, it is also weakly dominated. However, a weakly dominated strategy is not necessarily strictly dominated. ($s'<em>i \text{ is strictly dominated} \implies s'</em>i \text{ is weakly dominated} \not\impliedby$)

• Example 2.6: In Matrix 2.12, H weakly dominates M for Firm 1 because payoffs are equal for two of Firm 2's strategies, but H yields a strictly higher payoff for the third (5 > 2.5).

Iterated Deletion of Weakly Dominated Strategies (IDWDS)

• 2.7.1 Deletion Order Matters in IDWDS: This is a significant drawback. Unlike IDSDS, the order in which weakly dominated strategies are deleted can lead to different equilibrium predictions.

  • Example (Matrix 2.12):

    • Starting with Firm 1: Leads to IDWDS = ${(H, h), (H, m)}$.

    • Starting with Firm 2: Leads to IDWDS = ${(H, h), (M, h)}$.

  • The different equilibrium predictions demonstrate the path-dependency of IDWDS, limiting its general applicability.

• 2.7.2 IDSDS Vs. IDWDS:

  • The set of strategy profiles surviving IDWDS is always a subset of those surviving IDSDS. This means IDWDS generally has "more bite" (provides more precise predictions) by eliminating more strategies.

  • $s \text{ survives IDWDS} \implies s \text{ survives IDSDS} \not\impliedby$

  • However, IDWDS's dependence on the order of deletion makes it a less robust solution concept than IDSDS.

Strictly Dominant Strategies

• Context: Sometimes, IDSDS can eliminate all but one strategy for each player in the first round (e.g., Prisoner's Dilemma).

• Definition 2.5: Strictly Dominant Strategy: Player $i$ finds that strategy $s<em>i$ is strictly dominant if ui(si, s{-i}) > ui(s'i, s{-i}) for every strategy $s'</em>i \ne s<em>i$ and every strategy profile $s</em>{-i}$ of player $i$'s rivals.

  • Intuition: A strictly dominant strategy provides an unambiguously higher payoff than all other available strategies, regardless of opponents' actions.

• Definition 2.6: Strictly Dominant Equilibrium (SDE): A strategy profile $s^{SD} = (s^{SD}<em>i, s^{SD}</em>{-i})$ is an SDE if every player $i$ finds their strategy, $s^{SD}_i$, to be strictly dominant.

  • An SDE, if it exists, is the only strategy profile that survives IDSDS after just one round of deletion for each player.

• Example 2.7 (Finding SDEs):

  • Prisoner's Dilemma: "Confess" is a strictly dominant strategy for every player; thus (Confess, Confess) is an SDE.

  • Coordination and Anticoordination Games: Battle of the Sexes, Stag Hunt, and Game of Chicken do not have strictly dominant strategies, and therefore no SDE exists.

• Relationship to IDSDS: If a strategy is an SDE, it must also survive IDSDS. However, a strategy profile surviving IDSDS does not necessarily have to be an SDE (e.g., in Example 2.2, (H,h) survived IDSDS but H was not strictly dominant in the original game).

  • $s \text{ is an SDE} \implies s \text{ survives IDSDS} \not\impliedby$

Evaluating SDE as a Solution Concept

• 1. Existence? No. Many games lack strictly dominant strategies for all players, so an SDE may not exist.

• 2. Uniqueness? Yes. If an SDE exists in a game, it must be unique. (Proof by contradiction: assuming two SDEs would lead to a contradiction of the strict dominance definition).

• 3. Robust to Small Payoff Perturbations? Yes. Like IDSDS, the strict inequalities defining a strictly dominant strategy hold for infinitesimally small payoff changes.

• 4. Socially Optimal? No. The SDE of a game is not necessarily socially optimal (e.g., Prisoner's Dilemma's (Confess, Confess) is not Pareto optimal).

Applying IDSDS in Common Games

• 
  

• 2.4.1 Prisoner's Dilemma Game:

  • Scenario: Two prisoners, separated, face choices to Confess (C) or Not Confess (NC). Individual incentives lead to a collectively worse outcome.

  • 
    

  • Matrix 2.4a (Payoffs as negative years in jail):
    

    Player 2	Confess	Not confess
    Player 1 Confess	-4, -4	0, -8
    Not confess	-8, 0	-2, -2
    IDSDS Result: (Confess, Confess). Both players confess, serving 4 years each. This is an SDE.
    Implication: This outcome is not Pareto optimal, as (Not confess, Not confess) yields (-2,-2), making both players better off.
    General Form (Matrix 2.4d): Payoffs must satisfy a > c and b > d. If d > a, players could improve by coordinating on (Not confess, Not confess).
    Real-world Examples: Price wars, tariff wars, negative campaigning, and various movie plots.
    2.4.2 Coordination Games—The Battle of the Sexes Game:
    Scenario: A couple wants to be together but has conflicting preferences (Football or Opera). Both prefer to be at the same event.
    Matrix 2.5a:
    Wife	Football, F	Opera, O
    :---	:---:	:---:
    Husband Football, F	10, 8	6, 6
    Opera, O	4, 4	8, 10
    IDSDS Result: No strategies are strictly dominated for either player. All four strategy profiles survive IDSDS = ${(F, F), (F, O), (O, F), (O, O)}$. No SDE exists.
    Characteristic: Players have incentives to choose the same strategy as their opponent (positive network externality).
    General Form (Matrix 2.5b): Payoffs must satisfy ai > ci and di > bi for each player $i$. This structure ensures no strictly dominated strategies.
    2.4.3 Pareto Coordination Game—The Stag Hunt Game:
    Scenario: Two hunters can catch a small hare alone or hunt a larger stag together (which yields more meat). Both are better off coordinating on Stag.
    Matrix 2.6a:
    Player 2	Stag, S	Hare, H
    :---	:---:	:---:
    Player 1 Stag, S	6, 6	1, 4
    Hare, H	4, 1	2, 2
    IDSDS Result: No strictly dominated strategies. All four strategy profiles survive IDSDS = ${(S, S), (S, H), (H, S), (H, H)}$. No SDE exists.
    General Form (Matrix 2.6b): Payoffs must satisfy a > b \ge d > c. No strictly dominated strategies.
    2.4.4 Anticoordination Game—The Game of Chicken:
    Scenario: Two teenagers drive cars towards each other; swerving avoids a crash but makes one a "chicken." Staying makes one "top dog" but risking a crash if both stay.
    Matrix 2.7a:
    Player 2	Swerve	Stay
    :---	:---:	:---:
    Player 1 Swerve	-1, -1	-8, 10
    Stay	10, -8	-30, -30
    IDSDS Result: No strictly dominated strategies. All four strategy profiles survive IDSDS = ${(\text{Swerve, Swerve}), (\text{Swerve, Stay}), (\text{Stay, Swerve}), (\text{Stay, Stay})}$. No SDE exists.
    Characteristic: Players have incentives to choose a different strategy than their opponent (negative network externality or congestion games).
    General Form (Matrix 2.7b): Payoffs must satisfy c > d > b > a. This structure ensures no strictly dominated strategies.
    Symmetric and Asymmetric Games

    • Definition 2.2: Symmetric Game:

      • A two-player game is symmetric if:

        1. Both players' strategy sets coincide ($S<em>A = S</em>B$).

        2. Payoffs are unaffected by the identity of the player choosing each strategy, i.e., $u<em>A(s</em>A, s<em>B) = u</em>B(s<em>B, s</em>A)$ for every strategy profile $(s<em>A, s</em>B)$.

        3. Visually:

           • Same number of rows and columns, with same action labels.

           • Payoffs on the main diagonal coincide: $u<em>A(s,s) = u</em>B(s,s)$.

           • Cells above the main diagonal are mirror images of those below: $u<em>A(s</em>A,s<em>B) = u</em>B(s<em>B,s</em>A)$ when $s<em>A \ne s</em>B$.

      • Examples: Prisoner's Dilemma, Game of Chicken, and Matrix 2.1 are symmetric.

    • Definition 2.3: Asymmetric Game:

      • A two-player game is asymmetric if it violates at least one of the four properties of a symmetric game.

      • Example: The Battle of the Sexes game is asymmetric (e.g., $u<em>H(F,F)=10$ but $u</em>W(F,F)=8$, violating the main diagonal payoff coincidence).