Question

Consider a two-player game and suppose that \(s^{*}\) and \(t^{*}\) are Nash equilibrium strategy profiles in the game. Must it be the case that \(\left\\{s_{1}^{*}, t_{1}^{*}\right\\} \times\left\\{s_{2}^{*}, t_{2}^{*}\right\\}\) is a weakly congruous strategy set? Explain why or why not.

Short answer

No, it is not necessary. Mixed strategies may not form a weakly congruous set.

Step-by-step solution

Understanding the Problem

We are given two Nash equilibrium strategy profiles \(s^{*}\) and \(t^{*}\). We need to determine if the set \(\{s_1^{*}, t_1^{*}\} \times \{s_2^{*}, t_2^{*}\}\) forms a weakly congruous strategy set.

Definition of Nash Equilibrium

A Nash equilibrium is a strategy profile where no player can benefit by unilaterally deviating from their strategy, given the strategies of other players.

Definition of Weakly Congruous Strategy Set

A weakly congruous strategy set means that each strategy in the set is best for each player given the strategies of the others in the set, and there is some overlap or a set of common strategies.

Analyzing Strategy Profiles

The strategy profiles \(s^{*}\) and \(t^{*}\) are Nash equilibria, meaning that they are individually pairs \((s_1^{*}, s_2^{*})\) and \((t_1^{*}, t_2^{*})\) where neither player wants to deviate. This does not necessarily imply that the mix of these strategies forms another Nash equilibrium.

Constructing Strategy Set

Form the set \(\{s_1^{*}, t_1^{*}\} \times \{s_2^{*}, t_2^{*}\}\). The combination creates four pairs: \((s_1^{*}, s_2^{*})\), \((s_1^{*}, t_2^{*})\), \((t_1^{*}, s_2^{*})\), and \((t_1^{*}, t_2^{*})\). We need to check if these pairs are best responses or form another Nash equilibrium.

Checking Congruence and Overlap

Since \(s^{*}\) and \(t^{*}\) are Nash equilibria, \( (s_1^{*}, s_2^{*}) \) and \( (t_1^{*}, t_2^{*}) \) are best response pairs. However, the mixed pairs \( (s_1^{*}, t_2^{*}) \) and \( (t_1^{*}, s_2^{*}) \) may not be Nash equilibria, since they are not guaranteed to be best responses given the respective strategies from different equilibria.

Conclusion

It is not necessary that \(\{s_1^{*}, t_1^{*}\} \times \{s_2^{*}, t_2^{*}\}\) forms a weakly congruous strategy set because mixed strategies from different Nash equilibria might not be optimal responses and thus may not satisfy the requirement of a weakly congruous strategy set.

Key concepts

Nash Equilibrium

In game theory, a Nash Equilibrium represents a situation where each player's strategy is optimal given the strategies of all other players. In simpler words, it is the set of strategies where no player has anything to gain by changing only their strategy. This means, if you are playing a game and you are in a Nash Equilibrium, you can only do better if someone else changes their strategy too.
This concept can be better understood with an example. Imagine two companies setting prices for similar products. If both companies choose prices where neither can gain more customers by changing their price alone, they are in a Nash Equilibrium.
Some critical elements of Nash Equilibrium include:

• No unilateral benefit from changing strategies: Players don't profit from deviating without others changing their strategies.
• Equilibrium might not be efficient: Sometimes, the Nash Equilibrium does not lead to the most socially beneficial outcome.
• Multiple equilibria: A game might have more than one Nash Equilibrium, meaning there might be several strategies that could satisfy the Nash conditions.

Understanding Nash Equilibrium is crucial because it provides insight into predicting outcomes in strategic scenarios, ranging from economics to everyday decisions.

Strategy Profiles

In the realm of game theory, a strategy profile is essentially a complete account of all the strategies selected by the players involved in a game. Think of it as a snapshot that lays out what every player intends to do at a given moment.
For each player in the game, a strategy profile denotes:

• The specific plan of action chosen by a player to maximize their payoff.
• The actions selected by other players, which influence their own strategy decision.

Consider two chess players: one might choose an aggressive opening, while the other opts for a defensive stance. The combination of these choices together forms the strategy profile of that particular game state.
The significance of strategy profiles is immense in games such as chess, business negotiations, or even political debates, as they provide the strategies motivating players’ actions. When evaluating these profiles, players aim to anticipate the best possible outcomes based on others’ strategies, making it a dynamic system heavily influenced by mutual beliefs and expectations.
Combining knowledge of strategy profiles with the concept of Nash Equilibrium helps players know when their chosen strategies no longer require adjustment, hence achieving a stable state where all participants are optimizing their outcomes.

Weakly Congruous Strategy Set

A weakly congruous strategy set in a game is an interesting concept that entails the idea of strategy sets being weakly aligned with each other. It refers to groups of strategies where each strategy is the best response given the strategies within this set. Importantly, there needs to be some sort of overlap, establishing common elements that connect these strategies.
To break it down, in a weakly congruous strategy set:

• There is alignment: Strategies are attractive responses in the context of what the other players are doing within the set.
• Overlapping or common ground: Despite possibly coming from different Nash Equilibria, these strategies share certain synergies or compatibilities.

However, in practical terms, not all combined sections of Nash Equilibriums guarantee a weakly congruous strategy set, as evidenced in the original solution of the exercise.
While true this alignment suggests orderliness within the chaotic landscape of strategy profiles, finding such sets can be challenging depending upon the complexity of the game and diversity in equilibrium strategies, such as mixing pieces from different equilibria that might not exactly align. Understanding this can provide insights into the inherent nature of cooperative and competitive strategies in a game, allowing deeper strategic analysis.