Replicator Dynamics

Activity (20 minutes)

Goal. Build the replicator equation from a physical process and watch a population converge to a stable rest point, so that frequency-dependent selection is felt before the equation is written down.

The deck. There is a write-on deck at /decks/replicator-dynamics/main.pdf: the snowdrift payoffs printed, blank axes with \(x = 2/3\) already marked so the trajectory can be plotted round by round, and the replicator equation assembled from what the room saw.

The room is a population. Each student is one individual playing the snowdrift game: two drivers meet at a snowdrift blocking the road and each chooses to Dig or Stay. Clearing the drift is worth 4 to each driver; digging costs 2, shared if both dig. With actions ordered (Dig, Stay) the row player's payoffs are

\[ A = \begin{pmatrix} 3 & 2 \\ 4 & 0 \end{pmatrix} \]

so two diggers get 3 each, a lone digger gets 2 while the stayer free-rides for 4, and two stayers get 0.

Run it as follows:

Every student picks Dig or Stay and stands on the corresponding side of the room. Record the fraction \(x\) on the Dig side.
With the current split, compute each side's fitness: \(f_D = 3x + 2(1 - x)\) and \(f_S = 4x\). Read off which side is doing better.
Evolutionary step. Lob a few small bribentives to the higher-fitness side, then move a small fixed number of students (say two or three) from the lower-fitness side to the higher-fitness side. Record the new \(x\).
Repeat for several rounds, plotting \(x\) against the round on the board.

Rewarding the fitter side each round makes the pull towards it tangible and gives students a reason to switch. Whatever the starting split, the population settles near \(x = 2/3\), where \(f_D = f_S\).

Debrief. Draw out the equation from what they saw:

A strategy that is doing well but is rare grows only slowly, because few individuals play it to begin with; a strategy that is doing well and is common grows fast. The number that switch is proportional to how many already play it (\(x\)) times how far above average its fitness is (\(f - \phi\)). That is exactly \(\dot{x} = x(f - \phi)\).
Each round, with a small step, is one step of Euler's method on the replicator equation; shrinking the step traces the continuous trajectory. This is the link to the numerical integration reading.
Contrast briefly with Rock-Paper-Scissors, where the same dynamics never settle but cycle: a useful caution that a rest point is not always reached.

Discussion (20 minutes)

Work through the Replicator Dynamics chapter.

Discussion Point: After the definition of the replicator dynamics equation, ask how this differs from our example?

Discussion Point: After the definition of an ESS, ask whether the rest point \(x = 2/3\) from our snowdrift game is an ESS, and contrast it with the Rock-Paper-Scissors equilibrium, which is not.

Discussion Point: After the characterisation of the ESS theorem ask how we could use this to find the ESS for the replicator dynamics equation?

Answers for the deck

The payoffs. Two diggers get 3 each. A lone digger gets 2 while the stayer free-rides for 4, and two stayers get 0.

Fitnesses. \(f_D = 3x + 2(1 - x) = x + 2\) and \(f_S = 4x\). They are equal at \(x = 2/3\), which is where the room settles whatever split it starts from.

The equation. A strategy grows in proportion to how far above the average \(\phi = x f_D + (1 - x) f_S\) it is, and to how many already play it, giving \(\dot{x} = x(f_D - \phi)\). Here \(\phi = -3x^{2} + 6x\), so

\[ \dot{x} = x\left(3x^{2} - 5x + 2\right) = x(3x - 2)(x - 1). \]

Each round of the activity, with its small fixed step, is one step of Euler's method on this equation.

Rest points and direction. \(\dot{x} = 0\) at \(x = 0\), \(x = 2/3\) and \(x = 1\). For \(0 < x < 2/3\) all three factors give \(\dot{x} > 0\), and for \(2/3 < x < 1\) they give \(\dot{x} < 0\). So from any interior start the population moves to \(x = 2/3\).

Is it an ESS? Yes. It is stable under the dynamics, and a small group of Stayers invading a population at \(x = 2/3\) earns less than the residents, so they cannot spread. Contrast Rock-Paper-Scissors, whose interior equilibrium is not an ESS: the same dynamics cycle around it instead of settling.

From the activity to the exam answer

The activity above is written up as a marked exam question: Question 1 (the in-class activity) on the Replicator Dynamics page, with a full worked solution. Closing the loop here is the step that helps students who find exams hard: work through that question together, or set it as the immediate follow-up, so they see the game they just played turned into a full-mark answer.

Communications

General email templates to send before and after this class. Fill in the bracketed placeholders before sending.

Before class

Hi all,

A reminder that our next Game Theory class covers Replicator Dynamics.

All of the course materials, including the relevant chapter, are available
at https://vknight.org/gt/. It is worth skimming the chapter beforehand.

See you in class,
Vince

After class

Dear all,

Thanks for your work in today's class on Replicator Dynamics.

A recording is available here [RECORDING LINK] and on Learning Central.

All class resources are available at https://vknight.org/gt/.

Thanks,
Vince