Deriving Nash Equilibrium with Best Response Functions

Imagine two partners working together on a project. Should they coordinate their efforts, or does one person’s extra work make the other’s contribution less valuable? These questions lie at the heart of strategic decision-making, and game theory offers a powerful tool to answer them: the best response function. This mathematical concept helps us understand how rational players choose their strategies when their outcomes depend not just on their own actions, but on the actions of others.

Understanding best response functions 🔗

In game theory, a best response function identifies the optimal strategy for a player given the strategies chosen by other players. Think of it as your personal playbook that tells you the smartest move for every possible action your opponent might take. For games with continuous strategies-where players can choose any value within a range rather than just a few discrete options-best response functions become essential for finding equilibrium points.

Consider a simple business scenario: two competing coffee shops deciding how much to spend on advertising. Each shop’s optimal advertising budget depends on what the other shop spends. If your competitor increases their advertising, you might need to respond by adjusting your own budget. The best response function captures this relationship mathematically, showing your optimal spending level for each possible choice your competitor makes.

The mathematical foundation 🔗

A best response for player i to a strategy profile of other players is formally defined as a strategy that maximizes player i’s payoff, given what everyone else is doing. In mathematical terms, if we denote player i’s strategy as s_i and everyone else’s strategies as s_-i, then s_i is a best response if it yields a payoff at least as high as any other strategy player i could choose.

For continuous strategy games, we typically find best response functions by taking the first-order condition of the payoff function. This involves differentiating the payoff with respect to the player’s own strategy and setting it equal to zero. The resulting equation, when solved, gives us the best response function that maps each possible strategy combination of opponents to the player’s optimal response.

Synergistic relationships and upward-sloping responses 🔗

Not all strategic situations involve conflict. Sometimes, players benefit when others exert more effort-a phenomenon called synergy. Consider a marriage where both partners invest time and energy into their relationship. When one partner increases their effort, it often makes the other partner’s contributions more valuable, creating a positive feedback loop.

In such synergistic relationships, the best response functions are upward-sloping, meaning that as one player increases their action, the other player’s optimal response is also to increase their action. This upward slope is the hallmark of what economists call strategic complements. If you and your business partner are developing a product together, and your partner puts in extra hours refining the design, you might find it worthwhile to invest more time in marketing-because a better product makes your marketing efforts more effective.

A mathematical example of synergy 🔗

Suppose each person in a partnership chooses an effort level, and their payoff increases with both their own effort and their partner’s effort. The payoff might look something like: Payoff = a × (own effort) + b × (partner’s effort) + c × (own effort) × (partner’s effort), where the last term captures the synergistic effect. Taking the derivative with respect to own effort and solving gives a best response that increases as the partner’s effort increases-an upward-sloping reaction curve.

Finding Nash equilibrium at the intersection 🔗

The real power of best response functions emerges when we use them to find Nash equilibrium-the point where everyone is simultaneously playing their best response to everyone else. A Nash equilibrium occurs where the best response functions for all players intersect.

Think of it geometrically: if you graph player 1’s best response function and player 2’s best response function on the same diagram, with each player’s strategy on a different axis, the Nash equilibrium appears at the point where these curves cross. At this intersection, player 1 is choosing the optimal response to player 2’s strategy, and player 2 is simultaneously choosing the optimal response to player 1’s strategy. Neither player has an incentive to deviate unilaterally.

In the synergistic marriage example, solving the system of linear best response equations yields an equilibrium where both partners exert identical, positive levels of effort. This symmetric outcome reflects the mutual reinforcement built into their relationship-they’re both motivated to contribute because the other person’s contribution makes their own effort more worthwhile.

Solving the system algebraically 🔗

To find the Nash equilibrium algebraically, we substitute one best response function into the other. If player 1’s best response is a function of player 2’s action, and player 2’s best response is a function of player 1’s action, we can write player 1’s equilibrium action as their best response to player 2’s best response to player 1’s action. This creates a fixed point-a consistent set of strategies where everyone’s expectations about others’ behavior are confirmed.

Strategic complements versus strategic substitutes 🔗

The slope of best response functions reveals fundamental differences in how players’ actions relate to each other. Actions are strategic complements if an increase in one player’s action raises the marginal return of the other player’s action, which mathematically means the cross-partial derivative of the payoff function is positive. This produces positively sloped best response functions.

Conversely, actions are strategic substitutes when an increase in one player’s action decreases the marginal return of the other’s action-indicated by a negative cross-partial derivative. This results in negatively sloped reaction curves. The distinction has profound implications for equilibrium behavior and comparative statics.

Real-world examples of each type 🔗

Strategic complements appear in many coordination scenarios. When companies adopt compatible technologies, one firm’s adoption makes it more valuable for others to adopt the same technology-think of how the value of social media platforms increases as more users join. In pricing decisions for complementary products, if firms sell products in partly differentiated markets where demand increases with competitors’ prices, their pricing decisions exhibit strategic complementarity.

Strategic substitutes dominate competitive situations. In quantity competition between firms (Cournot competition), when one firm increases output, it drives down market price and reduces the other firm’s incentive to produce. The best response is to cut back production-quantities are strategic substitutes. Similarly, in advertising battles, if your competitor floods the market with ads, the return on your advertising might diminish, leading you to scale back rather than escalate.

The mathematical signature 🔗

The mathematical test for distinguishing these cases relies on the sign of the cross-partial derivative. For player i with payoff function π_i(a_i, a_j), we examine ∂²π_i / ∂a_i∂a_j. If this cross-partial is positive, player j’s action increases the marginal benefit of player i’s action-they’re strategic complements with an upward-sloping best response. If it’s negative, player j’s action decreases player i’s marginal benefit-they’re strategic substitutes with a downward-sloping best response.

This mathematical signature has practical implications. In games with strategic complements, multiple equilibria often exist because the mutual reinforcement can lead to coordination on either high or low levels of activity. In games with strategic substitutes, competitive forces typically push toward a unique equilibrium as players’ incentives to offset each other create stability.

What do you think? Can you identify situations in your own life or work where your optimal decisions depend on others’ choices? Are these situations characterized by strategic complements (where everyone benefits from coordination) or strategic substitutes (where you want to do the opposite of what others do)?

References

1. https://en.wikipedia.org/wiki/Best_response
2. https://eml.berkeley.edu/~webfac/card/e101a_s05/101duopolytheory.pdf
3. https://en.wikipedia.org/wiki/Strategic_complements