What does strictly dominated mean in game theory? This seemingly simple question opens a door to a fascinating area of strategic decision-making. Understanding strictly dominated strategies is crucial for analyzing games, predicting player behavior, and ultimately, achieving optimal outcomes. This exploration delves into the precise definition, identification methods, and real-world applications of this key concept within game theory, highlighting its significance in various strategic interactions.
We will examine how identifying and eliminating strictly dominated strategies can simplify complex games, revealing potential Nash Equilibria and offering valuable insights into rational player behavior. We will explore both simple 2×2 matrices and more complex scenarios, including iterative elimination and the implications for different game types. The discussion will also address the limitations of this concept, acknowledging situations where it may not provide a complete or accurate representation of strategic interactions.
Definition of Strictly Dominated Strategy
A strictly dominated strategy, in the context of game theory, represents a course of action for a player that always yields a worse outcome than another available strategy, regardless of the actions taken by other players. Understanding this concept is crucial for simplifying complex games and predicting rational player behavior.A player employing a strictly dominated strategy is essentially making a suboptimal choice.
Rational players, assuming they are aware of all available strategies and their payoffs, will always avoid strictly dominated strategies. This is because there’s always a better alternative available, irrespective of the actions of other players.
Formal Definition of a Strictly Dominated Strategy
Let’s consider a game with two players, Player 1 and Player 2. Player 1 has a set of strategies S1 = s 11, s 12, …, s 1n, and Player 2 has a set of strategies S2 = s 21, s 22, …, s 2m. The payoff for Player 1, given Player 1 chooses strategy s 1i and Player 2 chooses strategy s 2j, is denoted as u1(s 1i, s 2j) .
Similarly, the payoff for Player 2 is u2(s 1i, s 2j) .A strategy s 1k ∈ S1 is strictly dominated by another strategy s 1l ∈ S1 if and only if:
u1(s 1l, s 2j) > u 1(s 1k, s 2j) for all s2j ∈ S2
This means that for every possible action taken by Player 2, Player 1 receives a strictly higher payoff by choosing s 1l instead of s 1k. The same definition applies to Player 2, with the payoffs reversed.
Example of a Game with a Strictly Dominated Strategy
Consider a simple game where two players simultaneously choose either “Cooperate” (C) or “Defect” (D). The payoffs are represented in a payoff matrix:
| Player 2: C | Player 2: D | |
|---|---|---|
| Player 1: C | 3, 3 | 0, 5 |
| Player 1: D | 5, 0 | 1, 1 |
In this game, Player 1’s strategy “C” is strictly dominated by strategy “D”. Observe that if Player 2 chooses “C”, Player 1 gets 3 by cooperating and 5 by defecting. If Player 2 chooses “D”, Player 1 gets 0 by cooperating and 1 by defecting. In both cases, defecting yields a higher payoff for Player 1. Therefore, a rational Player 1 would always choose “D”, eliminating the strictly dominated strategy “C”.
Identifying Strictly Dominated Strategies
This section details the identification and analysis of strictly dominated strategies within the framework of game theory. We will explore various game matrix sizes, illustrating the process of identifying these strategies and their implications for game outcomes. The iterative elimination of strictly dominated strategies will also be examined.
2×2 Game Matrix Analysis
A 2×2 game matrix provides a straightforward setting to illustrate the concept of strictly dominated strategies. Analyzing these simple games helps build a foundation for understanding more complex scenarios.
2×2 Game Matrix Example with Strictly Dominated Strategies
The following 2×2 game matrix shows payoff values for Player 1 and Player 2, with strategies A and B for each player.
| Player 1 \ Player 2 | Strategy A | Strategy B |
|---|---|---|
| Strategy A | (2, 1) | (0, 3) |
| Strategy B | (3, 0) | (1, 2) |
For Player 1, Strategy B strictly dominates Strategy A. Regardless of Player 2’s choice, Player 1 receives a higher payoff by choosing Strategy B (3 > 2 and 1 > 0). For Player 2, neither strategy strictly dominates the other. If Player 1 chooses A, Player 2 prefers B (3 > 1). If Player 1 chooses B, Player 2 prefers A (0 > 2).
Therefore, Player 2 has no strictly dominated strategy.
2×2 Game Matrix Example without Strictly Dominated Strategies
In this example, neither player has a strictly dominated strategy.
| Player 1 \ Player 2 | Strategy A | Strategy B |
|---|---|---|
| Strategy A | (2, 2) | (0, 3) |
| Strategy B | (3, 0) | (1, 1) |
Player 1: If Player 2 chooses A, Player 1 prefers B (3 > 2). If Player 2 chooses B, Player 1 prefers A (0 > 1).Player 2: If Player 1 chooses A, Player 2 prefers B (3 > 2). If Player 1 chooses B, Player 2 prefers A (0 > 1).Neither player has a strategy that is always better than the other, regardless of the opponent’s choice.
Right, so strictly dominated in game theory means a strategy’s always a worse bet than another, regardless of what your opponents do. For a proper in-depth explanation and some cracking examples, check out the lighthouse knowledge base ; it’s got all the info you need. Basically, if one strategy consistently yields lower payoffs, it’s strictly dominated and a complete waste of time to consider it, innit?
3×3 Game Matrix Analysis
Extending the analysis to 3×3 matrices introduces greater complexity, but the fundamental principle of identifying strictly dominated strategies remains the same.
3×3 Game Matrix Example with Strictly Dominated Strategies and Iterative Elimination
The following 3×3 game matrix illustrates a scenario with strictly dominated strategies.
| Player 1 \ Player 2 | Strategy A | Strategy B | Strategy C |
|---|---|---|---|
| Strategy A | (1, 1) | (2, 3) | (3, 0) |
| Strategy B | (4, 0) | (3, 2) | (2, 1) |
| Strategy C | (0, 2) | (1, 4) | (0, 3) |
For Player 1, Strategy C is strictly dominated by Strategy B (4 > 0, 3 > 1, 2 > 0). Eliminating Strategy C results in the following 2×2 matrix:
| Player 1 \ Player 2 | Strategy A | Strategy B |
|---|---|---|
| Strategy A | (1, 1) | (2, 3) |
| Strategy B | (4, 0) | (3, 2) |
For Player 2, neither strategy is strictly dominated in the reduced matrix. No further iterative elimination is possible.
Comparing Strictly and Weakly Dominated Strategies
Strictly and weakly dominated strategies differ in the nature of the payoff comparison.
| Feature | Strictly Dominated Strategy | Weakly Dominated Strategy |
|---|---|---|
| Payoff Comparison | Always yields a lower payoff than another strategy, regardless of the opponent’s action. | Yields a lower payoff than another strategy in at least one scenario, and never a higher payoff. May yield equal payoffs in other scenarios. |
| Example | In the 2×2 matrix above (2.1a), Player 1’s Strategy A is strictly dominated. | Consider a modified matrix where Player 1’s Strategy A yields (1,1), (2,3), while Strategy B yields (2,0), (2,3). Strategy A is weakly dominated by B because it yields a lower payoff only when Player 2 chooses A. |
2×2 Game Matrix Example with a Weakly Dominated Strategy
The following matrix illustrates a weakly dominated strategy.
| Player 1 \ Player 2 | Strategy A | Strategy B |
|---|---|---|
| Strategy A | (1, 1) | (2, 3) |
| Strategy B | (2, 0) | (2, 3) |
Strategy A is weakly dominated by Strategy B for Player 1. If Player 2 chooses A, Player 1 gets a higher payoff with B (2 > 1). If Player 2 chooses B, Player 1 gets the same payoff with A and B (2 = 2). Because there is at least one case where B is better and no case where A is better, A is weakly dominated.
It is not strictly dominated because the payoffs are equal in one scenario.
Advanced Application: A Real-World Scenario
Consider a firm deciding whether to advertise (A) or not advertise (N) a new product, while a competitor simultaneously makes the same decision. Payoffs represent profit in millions of dollars.
| Firm 1 \ Firm 2 | Advertise (A) | Not Advertise (N) |
|---|---|---|
| Advertise (A) | (1, 1) | (3, 0) |
| Not Advertise (N) | (0, 3) | (2, 2) |
For Firm 1, “Not Advertise” is not a strictly dominated strategy. If Firm 2 advertises, Firm 1 earns more by advertising (1 > 0). If Firm 2 does not advertise, Firm 1 earns more by not advertising (3 > 2). Therefore, no strictly dominated strategy exists for either firm in this case. Note that this is a different example than what is described in the previous sections.
Iterated Elimination of Strictly Dominated Strategies
Iterated elimination of strictly dominated strategies is a solution concept in game theory that simplifies the analysis of complex games by sequentially removing strategies that are strictly dominated for each player. This process continues iteratively until no more strictly dominated strategies remain, potentially revealing a unique solution or reducing the game’s complexity to a more manageable size. The core idea rests on the assumption that rational players will never choose a strictly dominated strategy, as another strategy always yields a better payoff regardless of the other players’ actions.Iterated elimination of strictly dominated strategies involves the repeated application of the process of identifying and removing strictly dominated strategies.
The process is iterative because the removal of a strategy for one player can make a previously undominated strategy for another player now strictly dominated. This iterative process continues until no strictly dominated strategies remain for any player. The remaining strategies constitute the solution obtained through iterated elimination. The resulting game may have a single equilibrium or multiple equilibria, depending on the structure of the original game.
Iterated Elimination Process
The process begins by identifying any strictly dominated strategies for each player in the original game. These strategies are then removed from the game. The resulting reduced game is then examined for any new strictly dominated strategies that emerged due to the removal of the previous strategies. This process of identifying and removing strictly dominated strategies is repeated until no strictly dominated strategies remain.
The order of elimination can impact the final result if multiple strategies are simultaneously dominated.
Example of Iterated Elimination in a 3×3 Game
Consider the following 3×3 game matrix representing the payoffs for players A and B:
| B1 | B2 | B3 | |
|---|---|---|---|
| A1 | (2, 1) | (3, 2) | (1, 3) |
| A2 | (4, 0) | (1, 1) | (2, 2) |
| A3 | (0, 3) | (0, 0) | (3, 1) |
Step 1: For Player A, strategy A3 is strictly dominated by strategy A1. (A1 yields a higher payoff than A3 regardless of B’s choice). We eliminate A3.
| B1 | B2 | B3 | |
|---|---|---|---|
| A1 | (2, 1) | (3, 2) | (1, 3) |
| A2 | (4, 0) | (1, 1) | (2, 2) |
Step 2: For Player B, after removing A3, strategy B1 is strictly dominated by strategy B3. (B3 yields a higher payoff than B1 regardless of A’s choice between A1 and A2). We eliminate B1.
| B2 | B3 | |
|---|---|---|
| A1 | (3, 2) | (1, 3) |
| A2 | (1, 1) | (2, 2) |
Step 3: No further strictly dominated strategies exist. The iterated elimination process stops here. The remaining game is a 2×2 game.
Outcomes and Limitations
Iterated elimination of strictly dominated strategies can lead to a unique solution, representing a prediction of the game’s outcome under the assumption of rationality. However, it is crucial to acknowledge its limitations. The process is highly dependent on the order in which strategies are eliminated, and different elimination orders can lead to different outcomes. Furthermore, if no strictly dominated strategies exist, the method provides no solution.
It also fails to account for mixed strategies, where players randomize their actions. The method’s applicability is therefore limited to games where strictly dominated strategies exist and where the order of elimination does not affect the final outcome.
Strictly Dominated Strategies and Nash Equilibrium
Strictly dominated strategies play a crucial role in game theory, particularly in relation to Nash Equilibrium. Understanding their relationship simplifies game analysis and often leads to the identification of equilibrium points. This section will explore this relationship, demonstrating how the elimination of strictly dominated strategies can, under certain conditions, reveal Nash Equilibria.
Strictly Dominated Strategies and Best Responses
A strictly dominated strategy is one that always yields a lower payoff than another strategy, regardless of the actions of other players. Formally, for player i, strategy s i is strictly dominated by strategy s i‘ if for all possible strategy profiles of other players, the payoff from playing s i‘ is strictly greater than the payoff from playing s i.
This means a rational player will never choose a strictly dominated strategy because there’s always a better alternative. A best response is a strategy that maximizes a player’s payoff given the strategies chosen by other players. A strictly dominated strategy can never be a best response.Consider a simple 2×2 game represented by the following payoff matrix, where Player 1 chooses rows (A or B) and Player 2 chooses columns (C or D):Player 2| | C | D ||——-|——-|——-|| A | 2, 1 | 0, 0 || B | 3, 2 | 1, 1 |Player 1For Player 1, strategy A is strictly dominated by strategy B.
Regardless of whether Player 2 chooses C or D, Player 1 receives a higher payoff by choosing B (3 > 2 and 1 > 0). Therefore, Player 1 will rationally eliminate strategy A from consideration.
Iterated Elimination of Strictly Dominated Strategies and Nash Equilibrium
The iterative elimination of strictly dominated strategies is a process where strictly dominated strategies are sequentially removed from the game. This process simplifies the game, potentially revealing a Nash Equilibrium. A Nash Equilibrium is a set of strategies where no player can improve their payoff by unilaterally changing their strategy, given the strategies of other players. Example 1: Leading to a Unique Nash EquilibriumLet’s consider the following payoff matrix:Player 2| | C | D ||——-|——-|——-|| A | 2, 2 | 1, 0 || B | 3, 1 | 0, 0 |Player 1
1. Step 1
Right, so in game theory, a strictly dominated strategy means a player’s got a better option, no matter what the other player does. Think of it like this: you wouldn’t choose a rubbish strategy, would you? It’s a bit like searching for answers on a specific system, say the lenel knowledge base , instead of trawling the entire internet.
Basically, a strictly dominated strategy is just a massive waste of time; you’re always better off picking something else in the game.
For Player 1, strategy B strictly dominates A (3 > 2 and 0 > 1). Eliminate A.
2. Step 2
The simplified game becomes:Player 2| | C | D ||——-|——-|——-|| B | 3, 1 | 0, 0 |Player 1
3. Step 3
For Player 2, strategy C strictly dominates D (1 > 0). Eliminate D.
4. Step 4
The remaining game is (B, C) which is the unique Nash Equilibrium (3, 1). Example 2: Multiple Nash Equilibria RemainConsider this payoff matrix:Player 2| | C | D ||——-|——-|——-|| A | 2, 2 | 1, 1 || B | 1, 1 | 3, 3 |Player 1No strategy is strictly dominated for either player.
Therefore, iterative elimination of strictly dominated strategies does not simplify this game further. However, this game has two Nash Equilibria: (A, C) and (B, D).
Games Where Elimination Does Not Lead to a Unique Nash Equilibrium
Iterative elimination of strictly dominated strategies does not always lead to a unique Nash Equilibrium. The order of elimination can also matter in some cases. Here are three examples: Example 1: The game in Example 2 above demonstrates this. Example 2:Player 2| | C | D ||——-|——-|——-|| A | 2, 1 | 0, 0 || B | 1, 2 | 3, 1 |Player 1No strategy is strictly dominated, so the process cannot be applied.
However, there are multiple Nash Equilibria. Example 3:Player 2| | C | D ||——-|——-|——-|| A | 1, 1 | 3, 0 || B | 0, 3 | 2, 2 |Player 1No strategy is strictly dominated, leaving multiple Nash Equilibria.
Comparison of Games
| Game Matrix | Steps of Elimination | Resulting Nash Equilibria | Explanation |
|---|---|---|---|
| Example 1 (Unique NE) | A eliminated, then D | (B, C) | Unique NE found through sequential elimination. |
| Example 2 (Multiple NE) | No elimination possible | (A, C), (B, D) | No strictly dominated strategies; multiple NE remain. |
| Example 3 (Multiple NE) | No elimination possible | (A, C), (B, D) | No strictly dominated strategies; multiple NE remain. |
Limitations of Iterative Elimination
Iterative elimination of strictly dominated strategies is a powerful tool, but it has limitations. It cannot solve all games, particularly those without strictly dominated strategies. Moreover, the order of elimination can, in some cases, affect the outcome, leading to different results depending on which strategy is eliminated first. This is especially true in games with weakly dominated strategies (where one strategy is at least as good as another in all cases, and strictly better in at least one case). Many games have multiple Nash Equilibria, and this method will not identify them all if it cannot reduce the game to a single outcome.
Advanced Application: A 3×3 Game
Consider this 3×3 game:Player 2| | C | D | E ||——-|——-|——-|——-|| A | 1, 1 | 2, 0 | 0, 2 || B | 3, 2 | 1, 3 | 2, 1 || C | 0, 0 | 3, 1 | 1, 0 |Player 1
1. Step 1
For Player 1, strategy C is strictly dominated by B (3>0, 1>0, 2>1). Eliminate C.
2. Step 2
The simplified game is:Player 2| | C | D | E ||——-|——-|——-|——-|| A | 1, 1 | 2, 0 | 0, 2 || B | 3, 2 | 1, 3 | 2, 1 |Player 1
3. Step 3
For Player 2, strategy E is strictly dominated by D (3>2, 1>0). Eliminate E.
4. Step 4
The simplified game is:Player 2| | C | D ||——-|——-|——-|| A | 1, 1 | 2, 0 || B | 3, 2 | 1, 3 |Player 1
5. Step 5
No further elimination is possible. The Nash Equilibrium is (B, D) (1,3).
Real-World Application: Business Competition
Consider two competing coffee shops deciding on their pricing strategies. Each can choose a high price (H) or a low price (L). If both choose high prices, they each earn a moderate profit. If both choose low prices, they each earn a small profit. If one chooses a high price and the other a low price, the low-price shop earns a large profit, while the high-price shop earns almost nothing.
This can be modeled as a game where a high price strategy is strictly dominated for a rational player if the competitor chooses a low price. The elimination of strictly dominated strategies can help predict the likely outcome (a low-price equilibrium for both) under certain assumptions about player rationality.
Applications of Strictly Dominated Strategies
The concept of strictly dominated strategies, while seemingly straightforward, finds significant application across various fields, offering valuable insights into strategic decision-making and predicting outcomes in diverse scenarios. Understanding how to identify and eliminate these strategies can simplify complex games and lead to more effective decision-making. This section explores real-world examples, applications in different auction types, and analyses within various game structures to illustrate the practical relevance of this theoretical concept.
Real-world Examples of Strictly Dominated Strategies
The following table presents three distinct real-world examples where employing a strictly dominated strategy led to demonstrably negative outcomes. Analyzing these cases highlights the practical implications of failing to identify and avoid such strategies.
| Actors Involved | Available Strategies | Strictly Dominated Strategy | Outcome of Employing the Strictly Dominated Strategy | Outcome Had a Superior Strategy Been Chosen |
|---|---|---|---|---|
| A small business owner competing with a large corporation | (a) Price matching the corporation; (b) Maintaining higher prices and focusing on quality and customer service; (c) Engaging in a price war | (c) Engaging in a price war | Bankruptcy due to unsustainable pricing | Survival and potential growth by focusing on a niche market and superior customer service |
| A political candidate in a two-candidate race | (a) Focusing on core issues; (b) Engaging in negative campaigning; (c) Ignoring the opponent’s campaign | (c) Ignoring the opponent’s campaign | Significant loss of votes and electoral defeat | Improved chances of winning by addressing key issues and responding to the opponent’s campaign effectively |
| An investor in the stock market | (a) Diversifying investments; (b) Investing in a single high-risk stock; (c) Holding cash without investment | (c) Holding cash without investment (during a period of market growth) | Missed opportunities for significant returns | Substantial gains through diversified investments or strategic high-risk/high-reward ventures |
A Seemingly Strictly Dominated Strategy with a Positive Outcome, What does strictly dominated mean in game theory
Consider a situation where a company chooses to launch a product with demonstrably inferior technical specifications compared to competitors’ offerings. This strategy appears strictly dominated, as the superior products should capture the majority market share. However, if the company successfully positions its product as a significantly cheaper alternative, focusing on a price-sensitive market segment overlooked by competitors, the outcome might be positive.
The success hinges on factors beyond pure technical superiority – marketing, branding, and targeting a specific niche customer base. The standard definition of a strictly dominated strategy, which focuses solely on payoff values, fails to account for these external factors and the possibility of manipulating market dynamics. The company’s strategy succeeded not because of inherent product superiority but because of effective market segmentation and strategic pricing, demonstrating the limitations of a purely payoff-based analysis in predicting outcomes in complex real-world scenarios.
The success highlights the importance of considering contextual factors that the simplistic model of strictly dominated strategies might overlook. A successful outcome in this instance rests upon a careful consideration of the market, the competition, and the company’s ability to create demand despite the product’s technical inferiority.
Strictly Dominated Strategies in Auctions
In a sealed-bid, second-price auction, a bidder’s strategy is their bid amount. A strictly dominated strategy is a bid that yields a lower payoff than another bid, regardless of the other bidders’ actions. A rational bidder will never bid more than their true valuation of the item, as this is strictly dominated by a bid equal to their valuation.
If the bidder wins, they pay the second-highest bid, and if they lose, they pay nothing. Bidding above their valuation only increases their potential cost without increasing their probability of winning.Consider a numerical example with two bidders, A and B, whose valuations are $10 and $8 respectively. The payoff matrix is simplified, as the payoffs depend only on whether the bid is higher than the other bid.
| Bidder A | Bidder B | Payoff (A, B) |
|---|---|---|
| A bids > $8 | B bids ≤ $8 | ($10 – B’s bid, 0) |
| A bids > $8 | B bids > $8 | (0, $8 – A’s bid) |
| A bids ≤ $8 | B bids ≤ $8 | (0, 0) |
| A bids ≤ $8 | B bids > $8 | (0, $8 – A’s bid) |
A bid by A exceeding $10 is strictly dominated by a bid of $10. Similarly, any bid by B exceeding $8 is strictly dominated by a bid of $8.
Strictly Dominated Strategies in English vs. Dutch Auctions
Strictly dominated strategies are less prevalent in English auctions (open outcry) compared to Dutch auctions (descending price). In English auctions, the dynamic bidding process provides continuous information about other bidders’ valuations, making it difficult to employ a strictly dominated strategy without revealing information about one’s own valuation. In contrast, the sealed-bid nature of Dutch auctions creates information asymmetry. Bidders lack information about the other bidders’ valuations, increasing the possibility of inadvertently choosing a strictly dominated strategy.
The descending price structure in a Dutch auction also encourages bidders to bid earlier to avoid missing the opportunity, potentially leading to bids that are higher than their valuation.
Strictly Dominated Strategies in Different Game Types
Simultaneous games involve players making decisions concurrently without knowing the others’ choices. Consider a game with two players, each choosing either “Cooperate” (C) or “Defect” (D).
| Player 1 | Player 2 | Payoff (Player 1, Player 2) |
|---|---|---|
| C | C | (3, 3) |
| C | D | (0, 4) |
| D | C | (4, 0) |
| D | D | (1, 1) |
For Player 1, choosing “C” is not strictly dominated because it yields a higher payoff than “D” when Player 2 chooses “C”. However, if the payoff for (C,C) was (2,2), then C would be strictly dominated by D for both players.
Strictly Dominated Strategies in Sequential Games
In sequential games, players make decisions sequentially, with each player observing the previous player’s actions. Consider a game tree where Player 1 chooses between “Left” (L) or “Right” (R), and Player 2 then chooses between “Up” (U) or “Down” (D), depending on Player 1’s choice.[Imagine a game tree here: Player 1 chooses L or R. If L is chosen, the game branches into two sub-games with Player 2 choosing U or D.
If R is chosen, there is a similar branching with U or D options for Player 2. Payoffs are assigned to the terminal nodes of the game tree.]If, for example, the payoffs are such that regardless of Player 2’s choice, Player 1 always receives a higher payoff by choosing R, then L is a strictly dominated strategy for Player 1.
Eliminating this simplifies the game to only analyzing Player 1 choosing R, and the subsequent choices of Player 2.
Strictly Dominated Strategies in Zero-Sum vs. Non-Zero-Sum Games
Example of Zero-Sum Game with Strictly Dominated Strategy: Consider a simple matching pennies game. Player A chooses heads (H) or tails (T), and Player B does the same. If the choices match, Player A wins $1 from Player B; otherwise, Player B wins $1 from Player A. If Player A always chooses H, this is not strictly dominated, as it can win. However, if we modify the game so that if Player A chooses H and Player B chooses T, Player A loses $2 instead of winning $1, then choosing H becomes strictly dominated by choosing T for Player A.
Example of Non-Zero-Sum Game with Strictly Dominated Strategy: The Prisoner’s Dilemma is a classic example. Two suspects are arrested and interrogated separately. Each can either cooperate (remain silent) or defect (betray the other). If both cooperate, they receive a light sentence (e.g., 1 year). If both defect, they receive a moderate sentence (e.g., 5 years). If one defects and the other cooperates, the defector goes free while the cooperator receives a harsh sentence (e.g., 10 years). In this game, the strategy of cooperating is strictly dominated by the strategy of defecting for each player, as defecting always yields a better payoff regardless of the other player’s choice.
Strictly Dominated Strategies and Rationality
The concept of strictly dominated strategies is intrinsically linked to the assumption of rationality in game theory. Understanding how rational players interact with these strategies, and the scenarios where deviations from rationality occur, is crucial for applying game theory to real-world situations. This section explores the assumptions of rationality, the implications of strictly dominated strategies for rational players, and the various factors that can lead to deviations from perfectly rational behavior.
Assumptions of Rationality in Game Theory
Rationality, in game theory, refers to the assumption that players consistently strive to maximize their own payoff, given their beliefs about the other players’ actions. However, the precise interpretation of “rationality” can vary.
- Perfect Rationality: This assumes players possess complete information about the game, including the payoffs of all possible outcomes and the strategies available to all players. They also have unlimited computational ability to analyze all possible strategies and flawlessly execute the optimal one. Example: In a simple game of matching pennies, a perfectly rational player would always randomize their choice (heads or tails) with equal probability, preventing the opponent from exploiting any predictable pattern.
- Procedural Rationality: This focuses on the process of decision-making rather than the outcome. A procedurally rational player follows a consistent set of rules or heuristics to make decisions, even if those rules don’t guarantee the absolute best outcome. Example: A chess player might consistently employ a specific opening strategy, even if there are other openings that might statistically yield slightly better results on average.
The consistency of the process is valued.
- Epistemic Rationality: This emphasizes the consistency of beliefs and actions. An epistemically rational player holds beliefs that are justified by the available evidence and acts in accordance with those beliefs. Example: A player in a bargaining game might believe that the other player is highly risk-averse, and base their strategy on this belief, even if it’s not entirely accurate.
Their actions are consistent with their beliefs.
The rationality assumption, however, has limitations. Bounded rationality acknowledges that players have cognitive limitations, limited information, and time constraints, preventing them from perfectly optimizing their choices. Cognitive biases, such as confirmation bias (favoring information supporting pre-existing beliefs), anchoring bias (over-reliance on initial information), and the availability heuristic (overestimating easily recalled events), can further distort rational decision-making. Incomplete information about the game or other players’ strategies also introduces uncertainty and can lead to deviations from rationality.Comparing perfect and bounded rationality, consider a simple game where two players simultaneously choose between “Cooperate” (C) and “Defect” (D).
A perfectly rational player, knowing the payoff matrix, would always choose the strategy that maximizes their payoff regardless of the other player’s choice. A boundedly rational player might simplify the decision-making process, perhaps using a heuristic like “always cooperate,” even if it’s not the best strategy in all scenarios.
Strictly Dominated Strategies and Rational Player Behavior
A strictly dominated strategy is formally defined as a strategy that always yields a lower payoff than another strategy, regardless of the actions of other players. Mathematically, for player i, strategy si is strictly dominated by strategy s’i if:
ui(s’ i, s -i) > u i(s i, s -i) for all s-i
where ui is player i‘s payoff function, and s-i represents the strategies of all other players.Consider a 2×2 payoff matrix:| | Player 2: A | Player 2: B ||——-|————-|————-|| Player 1: X | 2, 1 | 0, 0 || Player 1: Y | 3, 2 | 1, 1 |In this example, Player 1’s strategy X is strictly dominated by strategy Y because Y always yields a higher payoff for Player 1, regardless of Player 2’s choice.Iterative elimination of strictly dominated strategies involves repeatedly removing strictly dominated strategies from the game until no more such strategies exist.
This process simplifies the analysis and can sometimes lead to a unique solution.Consider a 3×3 payoff matrix:| | Player 2: A | Player 2: B | Player 2: C ||——-|————-|————-|————-|| Player 1: X | 1, 1 | 2, 0 | 0, 2 || Player 1: Y | 3, 2 | 4, 3 | 2, 1 || Player 1: Z | 0, 0 | 1, 1 | 1, 0 |
1. Step 1
Strategy Z is strictly dominated by Y for Player 1. Remove Z.
2. Step 2
The matrix becomes:| | Player 2: A | Player 2: B | Player 2: C ||——-|————-|————-|————-|| Player 1: X | 1, 1 | 2, 0 | 0, 2 || Player 1: Y | 3, 2 | 4, 3 | 2, 1 |No strategy is strictly dominated.A rational player, aiming to maximize their expected utility, would never choose a strictly dominated strategy.
This is because another strategy guarantees a higher payoff regardless of the opponent’s actions. Therefore, choosing a strictly dominated strategy is irrational.The relationship between strictly dominated strategies and Nash Equilibrium is that all Nash Equilibria are free of strictly dominated strategies. However, the absence of strictly dominated strategies does not guarantee the existence of a Nash Equilibrium.
Deviations from Rationality
Several scenarios can lead to a rational player deviating from choosing a strictly dominated strategy.
- Misunderstanding of the Game: A player might incorrectly assess the payoffs or the strategies available, leading them to choose a dominated strategy due to a lack of complete information. This stems from informational limitations.
- Emotional Factors: Anger or revenge can override rational calculations, leading to the selection of a dominated strategy that punishes the opponent even if it means a lower payoff for oneself. For instance, a company might engage in a price war even if it’s economically damaging, simply to harm a competitor.
- Bounded Rationality and Heuristics: Players might use simple heuristics or rules of thumb that lead to suboptimal choices, especially in complex games where full optimization is computationally infeasible. This is a consequence of cognitive limitations.
Risk aversion and ambiguity aversion can also influence strategy choices. A risk-averse player might prefer a certain, smaller payoff over a potentially larger but riskier payoff, even if the riskier option is not strictly dominated in terms of expected value. Similarly, ambiguity aversion leads players to avoid strategies with uncertain payoffs, even if those uncertainties are favorable in expectation.The table provided in the prompt accurately summarizes factors leading to deviations from rationality.
Further Exploration
In repeated games, the possibility of future interactions can alter the incentives to choose a strictly dominated strategy. A player might choose a cooperative strategy in a repeated Prisoner’s Dilemma, even if defecting is strictly dominant in a single round, because cooperation can lead to higher payoffs over multiple rounds through establishing trust and reciprocity. The potential for future interactions changes the calculation of expected utility, making strictly dominated strategies potentially rational in a repeated game context.
Mixed Strategies and Strictly Dominated Strategies
A strictly dominated strategy, by definition, yields a lower payoff than another strategy regardless of the opponent’s actions. This raises the question of whether such a strategy could ever be part of a mixed strategy Nash Equilibrium, where players randomize their actions. The answer, surprisingly, is no. The inherent inferiority of a strictly dominated strategy renders it irrelevant in the calculation of optimal mixed strategies.A strictly dominated strategy will never be played with positive probability in a mixed strategy Nash Equilibrium.
This is because a player can always improve their expected payoff by replacing any probability assigned to a strictly dominated strategy with probability assigned to a dominating strategy. The inclusion of a strictly dominated strategy in a mixed strategy profile would imply a player is choosing an action that guarantees a lower payoff, contradicting the rationality assumption underlying Nash Equilibrium.
This has significant implications for game simplification and the analysis of equilibrium outcomes.
Strictly Dominated Strategies and Mixed Strategy Nash Equilibria
A strictly dominated strategy is never part of a mixed strategy Nash Equilibrium. Any mixed strategy assigning positive probability to a strictly dominated strategy can be improved upon by shifting that probability to the dominating strategy, thereby increasing the expected payoff. This holds true regardless of the opponent’s mixed strategy. Therefore, the process of iteratively eliminating strictly dominated strategies can simplify the analysis of games even when mixed strategies are considered, leading to a reduced game with the same Nash Equilibria in pure or mixed strategies.
Illustrative Game: The Interaction of Mixed and Strictly Dominated Strategies
Consider a two-player game with the following payoff matrix:
| Player 2 | ||
|---|---|---|
| Player 1 | Left | Right |
| Top | (1, 2) | (0, 0) |
| Bottom | (0, 1) | (2, 3) |
For Player 1, the strategy “Bottom” strictly dominates “Top.” Regardless of whether Player 2 chooses “Left” or “Right,” Player 1 receives a higher payoff by playing “Bottom.” Therefore, “Top” can be eliminated. The reduced game becomes:
| Player 2 | ||
|---|---|---|
| Player 1 | Left | Right |
| Bottom | (0, 1) | (2, 3) |
Even if Player 2 employs a mixed strategy, say playing “Left” with probability p and “Right” with probability (1-p), Player 1’s optimal strategy remains “Bottom.” The expected payoff for Player 1 from playing “Bottom” is always higher than that from playing “Top,” regardless of the value of p. This demonstrates that the strictly dominated strategy (“Top”) is never part of any Nash Equilibrium, even when mixed strategies are considered.
The unique Nash Equilibrium of the reduced game (and therefore the original game) is Player 1 playing “Bottom” and Player 2 playing “Right”.
Variations and Extensions: What Does Strictly Dominated Mean In Game Theory
The concept of strictly dominated strategies, while powerful, forms a foundation upon which more nuanced game-theoretic analyses are built. This section explores variations of dominance, related concepts, and comparisons with other solution concepts, enriching our understanding of strategic decision-making. These extensions address complexities not fully captured by the basic definition of strictly dominated strategies.
Variations of Strictly Dominated Strategies
Several variations and extensions build upon the core concept of strictly dominated strategies, enhancing its applicability to a wider range of game-theoretic scenarios. These variations offer a more refined understanding of strategic interactions and decision-making processes.
Weakly Dominated Strategies
A weakly dominated strategy is one that yields a payoff that is less than or equal to the payoff of another strategy, for all possible actions of the other players, and strictly less for at least one action profile. Unlike strictly dominated strategies, a weakly dominated strategy may yield the same payoff as another strategy in some circumstances. This subtle difference significantly alters its implications.
For instance, consider the following payoff matrix for players A and B:| | B1 | B2 ||——-|——-|——-|| A1 | 2, 2 | 1, 1 || A2 | 1, 1 | 3, 3 |Here, A1 is weakly dominated by A2.
A2 yields a strictly higher payoff when B plays B2 (3>1) and an equal payoff when B plays B1 (1=1). Unlike a strictly dominated strategy, a weakly dominated strategy is not always eliminated in rational decision-making. The choice depends on the players’ risk aversion and beliefs about the other player’s actions. A player might choose a weakly dominated strategy if they believe the probability of the situation where it yields a better outcome is high enough to offset the risk of lower payoffs in other situations.
This contrasts with strictly dominated strategies, which are always irrational to play.
Iterated Elimination of Dominated Strategies
Iterated elimination of dominated strategies involves sequentially removing dominated strategies from a game, simplifying the game structure and potentially revealing a unique solution. This process can be applied to both strictly and weakly dominated strategies, although the results may differ. Consider the following 3×3 game:| | B1 | B2 | B3 ||——-|——-|——-|——-|| A1 | 1, 1 | 2, 3 | 3, 2 || A2 | 2, 2 | 1, 1 | 4, 4 || A3 | 3, 3 | 4, 2 | 2, 1 |Step 1: A3 is strictly dominated by A1 (1>0, 2>0, 3>0) and A2 (1>0, 2>0, 3>0).
Removing A3 yields:| | B1 | B2 ||——-|——-|——-|| A1 | 1, 1 | 2, 3 || A2 | 2, 2 | 1, 1 |Step 2: B2 is strictly dominated by B1 for player B (2>1, 3>1). Removing B2 yields:| | B1 ||——-|——-|| A1 | 1, 1 || A2 | 2, 2 |Step 3: A2 strictly dominates A1.
The solution is (A2, B1).This process leads to a unique solution. However, the order of elimination can matter if only weakly dominated strategies are considered, and the process may not always lead to a unique solution.| Step | Action | Resulting Game ||—|—|—|| 1 | Eliminate A3 (strictly dominated) | 2×3 game || 2 | Eliminate B2 (strictly dominated) | 2×2 game || 3 | Eliminate A1 (strictly dominated) | (A2, B1) solution |
Dominated Strategies in Games with Mixed Strategies
The concept of dominated strategies extends to games with mixed strategies. A mixed strategy is dominated if there exists another mixed strategy that yields a weakly higher expected payoff regardless of the opponent’s strategy. Consider a game where Player A chooses between A1 and A2, and Player B chooses between B1 and B2:| | B1 | B2 ||——-|——-|——-|| A1 | 2, 1 | 0, 0 || A2 | 1, 2 | 1, 1 |Let’s consider a mixed strategy for A, denoted as pA1+(1-p)A2, where p is the probability of choosing A1.
If p=0.5, the expected payoff for A against B1 is 1.5 and against B2 is 0.5. A pure strategy A2 gives a constant payoff of 1, regardless of B’s choice. Since A2 yields a higher payoff than the mixed strategy (0.5, 1.5) in at least one case (against B2) and equal in another, the mixed strategy with p=0.5 is weakly dominated by A2.
Illustrative Examples
The Prisoner’s Dilemma serves as a classic illustration of game theory concepts, particularly the application of strictly dominated strategies. Understanding this game helps solidify the theoretical understanding of rational decision-making under conditions of strategic interaction. The dilemma highlights how individual rationality can lead to collectively suboptimal outcomes.The Prisoner’s Dilemma involves two suspects arrested for a crime. The police lack sufficient evidence for a major conviction unless at least one suspect confesses.
Each suspect is interrogated separately and offered the following deal: if one confesses and the other remains silent, the confessor goes free while the silent suspect receives a 10-year sentence. If both confess, they each receive a 5-year sentence. If both remain silent, they each receive a 1-year sentence for a minor charge.
Prisoner’s Dilemma and Strictly Dominated Strategies
The payoff matrix below summarizes the possible outcomes for Suspect A and Suspect B. Each cell represents the outcome (years in prison) for Suspect A, Suspect B respectively.
| Suspect B Confesses | Suspect B Remains Silent | |
|---|---|---|
| Suspect A Confesses | (5, 5) | (0, 10) |
| Suspect A Remains Silent | (10, 0) | (1, 1) |
Let’s analyze Suspect A’s perspective. If Suspect B confesses, Suspect A receives 5 years if they confess and 10 years if they remain silent. If Suspect B remains silent, Suspect A receives 0 years if they confess and 1 year if they remain silent. In both scenarios, confessing yields a better outcome for Suspect A than remaining silent.
Therefore, “remaining silent” is a strictly dominated strategy for Suspect A. The same logic applies to Suspect B; “remaining silent” is also a strictly dominated strategy for Suspect B.
- The Dilemma: Each suspect faces a choice between confessing and remaining silent. The outcome depends on the other suspect’s choice.
- Payoff Matrix: The game is represented by a payoff matrix showing the years of imprisonment for each suspect based on their choices.
- Strictly Dominated Strategy Identification: For each suspect, the strategy of “remaining silent” is strictly dominated by the strategy of “confessing,” as confessing always yields a better outcome regardless of the other suspect’s choice.
- Rational Choice and Suboptimal Outcome: Even though both suspects would be better off remaining silent (resulting in only 1 year each), rational self-interest leads them both to confess, resulting in a worse outcome (5 years each).
Illustrative Examples
The concept of strictly dominated strategies is best understood through contrasting examples. Games with strictly dominated strategies allow for simplification through iterative elimination, leading to a clearer understanding of the optimal strategies for players. The Matching Pennies game, however, presents a different dynamic, highlighting the limitations of this simplification method.The absence of strictly dominated strategies in Matching Pennies exemplifies a class of games where each action is contingent upon the opponent’s choice, rendering the concept of dominance irrelevant.
Understanding this difference clarifies the applicability and limitations of the concept of strictly dominated strategies in game theory.
Matching Pennies: Absence of Strictly Dominated Strategies
In the Matching Pennies game, two players simultaneously choose either “Heads” (H) or “Tails” (T). Player 1 wins if the choices match, while Player 2 wins if the choices differ. There is no strategy that is always better than another, regardless of the opponent’s action. If Player 1 chooses Heads, Player 2 can choose Tails to win.
If Player 1 chooses Tails, Player 2 can choose Heads to win. The same logic applies to Player 2’s choices. This interdependency prevents any strategy from being strictly dominated.
Comparison of Matching Pennies and Games with Strictly Dominated Strategies
Games with strictly dominated strategies, unlike Matching Pennies, exhibit a clear hierarchy of choices. One strategy consistently yields a better payoff than another, irrespective of the opponent’s actions. For instance, consider a game where Player 1 can choose A or B, and Player 2 can choose X or Y. If Player 1’s payoffs for (A,X), (A,Y), (B,X), (B,Y) are (2,1), (3,0), (1,2), (0,3) respectively, then strategy B is strictly dominated by strategy A for Player 1, because A always yields a higher payoff regardless of Player 2’s choice.
This is not the case in Matching Pennies. The absence of such a clear dominance relationship is the key difference.
Matching Pennies Game Representation
The Matching Pennies game can be represented using a payoff matrix:
| Player 2: Heads | Player 2: Tails | |
|---|---|---|
| Player 1: Heads | 1, -1 | -1, 1 |
| Player 1: Tails | -1, 1 | 1, -1 |
The entries represent the payoffs (Player 1, Player 2). A positive value indicates a win for Player 1, while a negative value indicates a win for Player 2. The symmetry of the matrix visually reinforces the absence of strictly dominated strategies; there’s no row or column that is uniformly better than another.
Limitations of Strictly Dominated Strategies
Strictly dominated strategies, while offering a seemingly straightforward approach to game theory problem-solving, possess limitations that restrict their applicability and can lead to inaccurate or suboptimal outcomes. Understanding these limitations is crucial for employing the concept effectively and avoiding misleading conclusions.
Identifying Specific Limitations
The effectiveness of eliminating strictly dominated strategies is significantly hampered in scenarios characterized by imperfect information or incomplete knowledge of opponent payoffs. The following limitations highlight these challenges.
- Incomplete Information about Payoffs: If a player lacks complete knowledge of the opponent’s payoff matrix, identifying a strictly dominated strategy becomes problematic. A strategy might appear dominated based on incomplete information, but reveal itself as rational once complete information is obtained.
- Imperfect Information about Actions: Uncertainty regarding the actions of other players can render the identification of strictly dominated strategies unreliable. A strategy might seem dominated under one set of assumptions about opponent behavior but not under others.
- Dynamic Games and Sequential Rationality: In sequential games, a strategy that appears strictly dominated at one stage might become rational later in the game, depending on the actions taken by other players in previous stages. This challenges the simple elimination process.
- Higher-Order Beliefs and Strategic Uncertainty: Players do not simply consider their opponents’ actions but also their beliefs about their opponents’ beliefs, and so on. This higher-order reasoning can invalidate the assumption underlying strictly dominated strategy elimination, where players are assumed to have common knowledge of the game.
- Risk Aversion and Non-Expected Utility Maximization: The concept of strictly dominated strategies assumes players are risk-neutral expected utility maximizers. However, if players are risk-averse or exhibit different preferences, a strategy might not be eliminated even if it appears strictly dominated under a risk-neutral framework.
Real-world examples where relying solely on eliminating strictly dominated strategies leads to suboptimal or incorrect outcomes include:
- Bidding in Auctions: A bidder might initially perceive a high bid as strictly dominated if they believe their competitors will always bid lower. However, if competitors are also risk-seeking or have a higher valuation of the item, the high bid might not be dominated and could yield a higher payoff.
- Military Strategy: A military strategy might appear strictly dominated if it assumes the enemy will always employ a specific tactic. However, if the enemy adapts its strategy, the initially dominated strategy could become unexpectedly effective.
- Negotiations: A negotiation strategy that involves making significant concessions might appear strictly dominated if one assumes the other party will always exploit such concessions. However, if the other party values a collaborative outcome, the concession strategy might lead to a better overall outcome than a more aggressive approach.
Limitations in Games with More Than Two Players
The complexity of games increases significantly with the addition of more players, affecting the applicability of strictly dominated strategies. Identifying strictly dominated strategies becomes computationally more demanding, and the potential for strategic interactions among multiple players increases the likelihood that strategies initially perceived as dominated might prove rational given the actions of others.
Consider a three-player game where each player can choose either “Cooperate” (C) or “Defect” (D). Payoffs are as follows (Player 1’s payoff is listed first, followed by Player 2’s and Player 3’s payoffs):
(C, C, C): (3, 3, 3)
(C, C, D): (0, 0, 5)
(C, D, C): (0, 0, 5)
(C, D, D): (-1, -1, 7)
(D, C, C): (5, 0, 0)
(D, C, D): (2, 2, 2)
(D, D, C): (2, 2, 2)
(D, D, D): (1, 1, 1)
In this game, no strategy is strictly dominated for a single player in isolation. However, the strategic interdependence between the players makes it difficult to eliminate any strategy based solely on the concept of strict dominance. The outcome depends heavily on the beliefs and actions of all three players.
Situations of Inapplicability
The elimination of strictly dominated strategies does not always lead to a unique solution.
Consider the following 2×2 game matrix:
Player 2
| Left | Right | |
|---|---|---|
| Player 1 Top | (2, 1) | (0, 0) |
| Player 1 Bottom | (1, 2) | (1, 1) |
Neither player has a strictly dominated strategy. The elimination of strictly dominated strategies fails to provide a unique solution. Alternative solution concepts, such as Nash Equilibrium, would be necessary to analyze this game.
Applicability Across Different Information Structures
The following table compares the applicability of strictly dominated strategies in games with perfect, imperfect, and incomplete information:
| Criterion | Perfect Information | Imperfect Information | Incomplete Information |
|---|---|---|---|
| Predictability of Outcomes | High, given rationality | Lower, due to hidden actions | Lowest, due to unknown payoffs and actions |
| Ease of Identifying Strictly Dominated Strategies | Relatively easy | More challenging | Highly challenging or impossible |
| Reliability of Eliminating Strictly Dominated Strategies | High | Lower, potential for errors | Very low, potentially misleading |
Potential for Misinterpretations
Mixed strategies and cognitive biases can significantly affect the identification and elimination of strictly dominated strategies.
Mixed strategies introduce probabilistic elements into the players’ choices. A strategy might appear strictly dominated when considering only pure strategies, but it might become rational when mixed strategies are allowed. For example, consider a game where Player 1 chooses between Top and Bottom, and Player 2 chooses between Left and Right:
Player 2
| Left | Right | |
|---|---|---|
| Player 1 Top | (1, 1) | (0, 2) |
| Player 1 Bottom | (2, 0) | (1, 1) |
If Player 1 plays Top with probability p and Bottom with probability 1-p, and Player 2 plays Left with probability q and Right with probability 1-q, then Bottom is not strictly dominated for certain values of p and q.
Unforeseen circumstances can also render a seemingly strictly dominated strategy rational. For instance, a company might initially perceive a marketing campaign as strictly dominated due to high costs. However, if a competitor unexpectedly exits the market, the campaign might become a rational choice.
Cognitive biases such as confirmation bias (seeking information confirming pre-existing beliefs) and anchoring bias (over-relying on the first piece of information received) can lead to misinterpretations in identifying strictly dominated strategies. A player might overlook a strategy that is not strictly dominated due to these biases.
Advanced Analysis
In multi-stage games, the elimination of strictly dominated strategies must be performed sequentially, considering the game tree and the information available at each stage. A strategy that appears dominated at an early stage might become rational later, depending on the actions taken in previous stages. For example, consider a three-stage game of sequential moves where each player has two options, A and B.
Stage 1: Player 1 chooses A or B.
Stage 2: Player 2 observes Player 1’s choice and chooses A or B.
Stage 3: Player 1 observes Player 2’s choice and chooses A or B.
The payoffs are determined by the sequence of choices. The elimination of strictly dominated strategies must proceed backward from the final stage, considering the information available at each decision point. A strategy that seems dominated in Stage 1 might become rational given the potential responses in later stages.
Flowchart for Identifying and Eliminating Strictly Dominated Strategies
A flowchart illustrating the process would involve:
- Start: Define the game’s players, strategies, and payoff matrix.
- Identify Strictly Dominated Strategies: For each player, compare each strategy against all other strategies. If one strategy yields strictly lower payoffs than another regardless of the opponent’s actions, it is strictly dominated.
- Eliminate Strictly Dominated Strategies: Remove the strictly dominated strategies from the game matrix.
- Iterate: Repeat steps 2 and 3 until no more strictly dominated strategies can be identified.
- Solution? If a unique solution remains, the process is complete. If multiple solutions or no solution remains, the method is inconclusive, and other solution concepts should be applied.
- End
The flowchart would include decision points indicating whether strictly dominated strategies exist and potential pitfalls, such as the incomplete information mentioned previously, leading to inconclusive results.
Advanced Applications
The concept of strictly dominated strategies, while seemingly simple, finds surprisingly sophisticated applications within more complex game-theoretic frameworks. Its power lies in its ability to simplify analyses, even in scenarios involving repeated interactions, uncertainty, or behavioral complexities. This section explores these advanced applications.The iterative elimination of strictly dominated strategies provides a powerful tool for simplifying complex games, even those with incomplete information or multiple stages.
Its application in such settings often reveals hidden strategic implications not readily apparent through other analytical methods. Furthermore, understanding how strictly dominated strategies interact with other game-theoretic concepts enhances our ability to model and predict real-world strategic interactions.
Repeated Games
In repeated games, players interact multiple times. The possibility of future interactions significantly alters strategic considerations. A strategy that is strictly dominated in a single-period game might not be strictly dominated in a repeated game, as its long-term consequences could outweigh its immediate disadvantage. For example, consider a prisoner’s dilemma played repeatedly. While defecting is a strictly dominant strategy in a single round, cooperation might emerge as a Nash equilibrium in a repeated game if players employ strategies like “tit-for-tat,” which punishes defection in subsequent rounds.
The iterative elimination of strictly dominated strategies can still be applied, but it must consider the entire sequence of interactions and potential payoffs across all periods. This requires analyzing the game’s structure and the players’ beliefs about their opponents’ strategies throughout the repeated interactions.
Games with Incomplete Information
Games with incomplete information involve uncertainty about the other players’ characteristics, payoffs, or even the rules of the game itself. Strictly dominated strategies can still be identified and eliminated, but the process becomes more nuanced. The analysis must account for the players’ beliefs about the possible states of the world and their opponents’ private information. Bayesian games, which explicitly model players’ beliefs, are often used to analyze such scenarios.
Identifying strictly dominated strategies in Bayesian games requires evaluating expected payoffs, taking into account the probabilities assigned to different possible states of the world. This process can simplify the game significantly, allowing for a more tractable analysis of the equilibrium strategies. Consider a game where Player 1 doesn’t know Player 2’s type (e.g., aggressive or cooperative). If a strategy is strictly dominated for both types of Player 2, it can be eliminated regardless of Player 1’s beliefs.
Behavioral Game Theory
Behavioral game theory examines how actual human behavior deviates from the predictions of traditional game theory. Strictly dominated strategies, while theoretically irrational to play, are sometimes observed in experiments. This highlights the limitations of assuming perfect rationality. Understanding why players might choose strictly dominated strategies can offer insights into cognitive biases, bounded rationality, or social preferences. For instance, loss aversion might lead a player to choose a strategy that is strictly dominated in terms of expected monetary payoff but minimizes potential losses, even if it results in a lower overall expected gain.
Analyzing the frequency and context of such choices can improve the predictive power of game theory models by incorporating more realistic assumptions about human decision-making. Experimental studies often reveal systematic deviations from the predictions of rational choice theory, including the use of strictly dominated strategies, offering valuable data for refining behavioral game theory models.
Conceptual Diagrams
Visual representations are crucial for understanding the often abstract concepts within game theory. Diagrams provide a clear and concise way to illustrate the process of identifying and eliminating strictly dominated strategies, highlighting the logic and implications involved. The following sections detail several diagrammatic approaches to representing this process, each offering a unique perspective on the strategic interactions.
Flowchart for Identifying and Eliminating Strictly Dominated Strategies
A flowchart provides a step-by-step visual guide to the iterative process of eliminating strictly dominated strategies. This is particularly useful for 2×2 games, where the comparisons are relatively straightforward. The flowchart below depicts the process:[Imagine a flowchart here. The flowchart would begin with a rectangle labeled “Start with a 2×2 Game Matrix”. This would lead to a diamond labeled “Does any player have a strictly dominated strategy?”.
A “Yes” branch would lead to a rectangle labeled “Identify and eliminate the strictly dominated strategy”, resulting in a smaller matrix. A “No” branch would lead to a rectangle labeled “No strictly dominated strategies remain”. The “Identify and eliminate” rectangle would loop back to the diamond. The “No strictly dominated strategies remain” rectangle would lead to a rectangle labeled “End”.
Each rectangle and diamond would be clearly labeled with the relevant information. The process would show how the payoff matrix changes with each iteration of strategy elimination. The comparison of payoffs would be implicitly shown through the decision-making process within the diamond.]
Decision Tree for Strictly Dominated Strategies
Decision trees offer a different perspective, focusing on the sequential nature of choices and their resulting payoffs. This approach is particularly helpful in visualizing the consequences of choosing a strictly dominated strategy in a multi-player game. Consider a simplified 3-player game where each player has two choices, A and B.[Imagine a decision tree here. The tree would start with Player 1’s decision node, branching into A and B.
Each branch would lead to Player 2’s decision node, again branching into A and B. This would continue for Player 3. Each terminal node would show the payoff vector (Player 1, Player 2, Player 3). Branches leading to outcomes where strictly dominated strategies are played would be clearly highlighted. For example, if Player 1’s strategy A is strictly dominated, all branches stemming from Player 1 choosing A would be visually distinct, indicating inferior outcomes compared to branches stemming from Player 1 choosing B.]
Concept Map for Strictly Dominated Strategies
A concept map provides a visual representation of the relationships between strictly dominated strategies and other key concepts in game theory. This helps to situate the concept within a broader theoretical framework. The following table Artikels a possible concept map:
| Concept | Definition | Relationship to Strictly Dominated Strategies |
|---|---|---|
| Strictly Dominated Strategy | A strategy that always yields a lower payoff than another strategy, regardless of the actions of other players. | Central concept; the focus of the analysis. |
| Nash Equilibrium | A set of strategies where no player can improve their payoff by unilaterally changing their strategy. | Eliminating strictly dominated strategies can simplify the game and lead to a Nash Equilibrium. |
| Iterated Elimination of Dominated Strategies | A process of sequentially removing strictly dominated strategies until no more can be eliminated. | The method used to find solutions in games where strictly dominated strategies exist. |
| Rationality | The assumption that players will always choose strategies that maximize their own payoff. | Rational players will never choose a strictly dominated strategy. |
| Payoff Matrix | A table that shows the payoffs for each player for every possible combination of strategies. | The foundation upon which strictly dominated strategies are identified. |
Mathematical Formulation
The concept of strictly dominated strategies can be rigorously defined using mathematical notation, allowing for precise analysis and the development of formal proofs related to game theory. This formalization clarifies the conditions under which a strategy is strictly dominated and facilitates the application of mathematical techniques to solve games.The mathematical formulation relies on representing games in normal form. This involves defining the players, their strategy sets, and their payoff functions.
Formal Definition of Strictly Dominated Strategies
Consider a game with n players, denoted as 1, 2, …, n. Each player i has a finite set of pure strategies, denoted as Si. The payoff function for player i is denoted as ui(s1, s2, …, sn), where si ∈ Si represents the strategy chosen by player i, and (s1, s2, …, sn) represents the strategy profile of all players.
A strategy si ∈ Si for player i is strictly dominated if there exists another strategy si’ ∈ Si such that:
ui(si’, s-i) > ui(si, s-i) for all s-i ∈ S-i
where s-i represents the strategy profile of all players except player i, and S-i = S1 × S2 × … × Si-1 × Si+1 × … × Sn is the Cartesian product of the strategy sets of all players except player i. This inequality indicates that the payoff for player i is strictly higher when choosing si’ compared to si, regardless of the strategies chosen by the other players.
Mathematical Notation and Symbols
The following notation is commonly used:* n: Number of players.
i
Index representing a specific player (i ∈ 1, 2, …, n).
Si
Set of pure strategies available to player i.
si
A specific strategy chosen by player i (si ∈ Si).
s-i
Strategy profile of all players except player i.
S-i
Set of all possible strategy profiles for players other than i.
ui(s1, s2, …, sn)
Payoff function for player i, representing the payoff received given the strategy profile (s1, s2, …, sn).
∀
The universal quantifier, meaning “for all”.
Proof of the Inefficiency of a Strictly Dominated Strategy
Suppose strategy si is strictly dominated by si’. This means that ∀s-i ∈ S-i, ui(si’, s-i) > ui(si, s-i). A rational player will always prefer a higher payoff to a lower payoff. Therefore, a rational player will never choose a strictly dominated strategy si because there always exists another strategy si’ that yields a strictly higher payoff, irrespective of the actions of other players.
This demonstrates the inherent inefficiency of playing a strictly dominated strategy.
FAQ Corner
Can a strictly dominated strategy ever be a best response?
No. By definition, a strictly dominated strategy always yields a lower payoff than another strategy, regardless of the opponent’s actions. Therefore, it can never be a best response.
What’s the difference between strictly and weakly dominated strategies?
A strictly dominated strategy is always worse than another strategy. A weakly dominated strategy is at least as good as another strategy in some cases, but worse in others.
Are all Nash Equilibria free of strictly dominated strategies?
Yes. A rational player would never choose a strictly dominated strategy, so a Nash Equilibrium, which represents a stable state where no player has an incentive to deviate, cannot contain one.
How does incomplete information affect the identification of strictly dominated strategies?
Incomplete information can make it difficult or impossible to identify strictly dominated strategies definitively, as the payoffs associated with different actions may be uncertain.
