Introspection Dynamics

Motivating Example: Two firms learning through introspection

Consider a market with two competing firms.

  • Firm 1 can choose between Advertising (\(A\)) and Research (\(R\)).

  • Firm 2 has a larger range of strategic options: it can focus on Price Cut (\(P\)), Branding (\(B\)), or Innovation (\(I\)).

Each firm earns profits depending on the combination of actions they take:

  • When Firm 1 advertises and Firm 2 cuts prices, both gain some market share, but profits are modest.

  • When Firm 1 invests in research and Firm 2 chooses innovation, both achieve good long-term results.

  • Branding by Firm 2 is mainly effective when Firm 1 advertises.

  • Price cuts by Firm 2 are costly if Firm 1 is focusing on research, and so on.

These interactions can be represented by the two asymmetric payoff matrices, \(A\) for Firm 1 and \(B\) for Firm 2:

\[\begin{split}A = \begin{pmatrix} 3 & 5 & 4 \\ 4 & 2 & 6 \end{pmatrix}, \qquad B = \begin{pmatrix} 3 & 5 & 2\\ 1 & 4 & 6 \end{pmatrix}\end{split}\]

Given these asymmetric payoffs, how will the players’ strategies evolve over time if both adapt introspectively?

The Introspection Dynamics

First defined in [Couto2022] introspection dynamics on 2 player games is defined as a process on \(N=2\) individuals which can use \(M_1\) and math:M_2 actions respectively.

The process is defined as follows, at each step:

  1. One of the two player is selected to reconsider their action choice.

  2. The chosen player \(i\in\{1, 2\}\) randomly selects one of their other \(M_i - 1\) actions.

  3. The chosen player compares their payoff \(\pi\) with their currently assigned action to \(\tilde \pi\) the payoff they would have had with the alternative action. They compute \(\Delta=\tilde \pi - \pi\).

  4. They change their action to the new action with probability given by:

    \[\frac{1}{1 + e^{-\beta \Delta}}\]

\(\beta\) can be interpreted as a learning rate: if \(\beta\) is zero than the probability of picking the new action is uniformly random. A high value of \(\beta\) indicates that the play will choose the better action with certainty.

This process corresponds to a Markov chain (details of which can be found in [Couto2022]) which defines the steady state probability vector \(v\) which gives the probability of being in any given state of action pairs.

Using Nashpy

See Use Introspection Dynamics for guidance of how to use Nashpy to obtain numerical simulations of the Introspection dynamics process.