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Supplemental educational materials for tutorial articles in Interface Focus

This page contains original teaching materials and media curated from the web to accompany a pair of tutorial articles in the journal Interface Focus. The articles are available as part of a special issue on Game Theory and Cancer. The open-access costs for these articles have been generously provided by the Princeton Physical Sciences-Oncology Center, so both tutorials are available under a CC-BY 3.0 license. Skip to the bottom of this page for supplementary materials for these articles.

  1. Wu A, Liao D, Tlsty T D, Sturm J C, and Austin R H (2014). Game theory in the death galaxy: interaction of cancer and stromal cells in tumour microenvironment. Interface Focus 4, 20140028 (In press). PMC Journal- In Process.
  2. Liao D and Tlsty T D (2014). Evolutionary game theory for physical and biological scientists: I. Training and validating population dynamics equations. Interface Focus 4, 20140037 (In press). PMC Journal- In Process. (open-access online)
  3. Liao D and Tlsty T D (2014). Evolutionary game theory for physical and biological scientists: II. Population dynamics equations can be associated with interpretations. Interface Focus 4, 20140038 (In press). PMC Journal- In Process. (open-access online)

Tour: Evolutionary game theory for physical and biological scientists

This is a 3 hour 30 minute tutorial workshop for beginners interested in evolutionary game theory (EGT) models. Small teams (2-3 students) can use this page as a self-guided tour. If you would like to schedule a live guided tour (online or in person), please contact David Liao (priority for NCI Physical Sciences-Oncology Network investigators and patient advocates). Simulations are Java applets that run directly off this website. You need not install MatLab.

Benefits: After using this workshop, biological scientists should be comfortable reading literature about simple evolutionary game theory models (literature samples included here). Students will likely also be comfortable performing some basic modeling on their own (i.e. studying how changing matrix elements describing pairwise interactions changes population dynamics).

Limitations: It is important to be aware that this is neither a comprehensive review of findings nor a comprehensive review of methods from evolutionary game theory research. As with the rest of this website, this workshop is in α (alpha). There are multiple ways to seek out in-depth assistance or to have modeling done for you. You could look through the conversations at the evolutionary game theory google community. You could also try to contact authors listed in the references on this page.

Team ice-breakers | Back to top

What is a social interaction? (15 minutes) | Back to top

  1. Write down a list of 5 biological systems that display social interaction. This might be easier if at least one of you is a biological scientist.
  2. The preceding list answers the question, "What are some examples of social interactions?" As a second question, work with your partner to think conceptually about what constitutes a social interaction. Consider the phrases "choice-making," "decision," and "contextual consequences."
  3. What do you think constitutes evolutionary game theory?

Interactive population dynamics simulation (20 minutes) | Back to top

This browser doesn`t support Java 2 SDK, Standard Edition v1.3 for APPLETS

Open Jens Langer's differential equation solver by clicking the "Start DESSolver Applet" button to the right. For more information about the applet, see Dr. Langer's applet information page. Dr. Langer has released the applet under the GNU GPL license.

The example screen capture highlights two regions of the applet. Enter the following pair of expressions into the "Equations" box. If you are running into problems copying-and-pasting equations into the Java applet, this blog post by Kyle Hatlestad (Oracle) might be useful. 

(1 + 1*y1/(y1+y2) -2*y2/(y1+y2))*y1,1
(1 + 2*y1/(y1+y2) +0.5*y2/(y1+y2))*y2,1
The first line specifies dy1/dt, and the second line specifies dy2/dt. The ",1" appended to each line is an initial condition. Set the range of time, x, from 0 to 2 and the range of the dependent variable, y, from 0 to 5.

  1. The parameters are typeset in bold font in the pair of expressions above. Choose two of these parameters to nudge (e.g. increase by 50% or decrease by 50%). Can you guess how the y1 and y2 curves will be affected before you make the changes?
  2. Call y1 and y2 subpopulations 1 and 2, respectively. Both differential equations are organized into a quantity in parentheses, for example this is (1 + 1*y1/(y1+y2) -2*y2/(y1+y2)) in the first equation, multiplied by a subpopulation size, here y1. The big quantity in parentheses is called the fitness. Write down a short response (no more than 3 sentences) defining fitness in plain English.
  3. The ratio y1/(y1+y2) is sometimes called a frequency. "Frequency" can be used in statistical and temporal contexts. Please explain why the ratio y1/(y1+y2) is called a frequency in each sense (no more than 3 sentences for each sense).

Introduction to evolutionary game theory and game theory (45 minutes) | Back to top

Read this multiple-choice prompt with a partner: Evolutionary game theory is an example of a modeling framework used in the NCI Physical Sciences-Oncology Network to understand social aspects of biological systems. What statement about evolutionary game theory is best?

  1. Evolutionary game theory refers to a collection of mathematical models in which organisms (e.g. cells) are modeled as automated, robotic replicators. The propensity with which a replicator generates progeny is modified by the time-frequency with which it encounters other replicators (e.g. through pairwise interactions).
  2. In the prisoner’s dilemma, defector population share and per capita fitness both increase. However, average fitness decreases because interaction between defectors and cooperators decreases the per capita fitness of cooperators more than enough to cancel out the fitness increase of defectors.
  3. In normal tissue microenvironments, individual cells typically express rational behaviors (i.e. protecting long-term survival of the host). Following carcinogenesis, however, cells can also display irrational behaviors, such as circumvention of stress-related barriers. Because cells can now choose between two classes of strategies, it becomes necessary to employ evolutionary game theory to determine how individual cells might make these choices and alter overall tissue system dynamics.
  4. John Maynard Smith understood the evolution of socially interacting biological individuals by describing the possible behaviors that individual agents could choose. The notion of choice in this context refers to the essential assumption that each individual organism expresses a strategy that maximizes its payoff, as computed using a payoff matrix. A startling result is the applicability of this kind of modeling, not just to organisms that perform advanced cognition, e.g. hawks and doves, but also, to individual cells, for which evolution had long been thought to proceed primarily in a Darwinian fashion.
  5. In traditional models of population dynamics, cell numbers vary as though cells individually solved for anti-derivatives and then adjusted their behavior to obey governing equations. In evolutionary game theory, individual cells, instead, perform sophisticated computations using payoff matrices to decide among strategies so as to realize Nash equilibria.

Watch a video (16 minutes) | Back to top

This video describes a population dynamics model in which cell-cell collisions affect replication rates

Watch a video (13 minutes) | Back to top

This video describes tabular game theory and similarities with the population dynamics model in the previous video

Work with a partner to address the multiple-choice prompt from the beginning of this section.

Break (5 minutes)

Connecting EGT and GT with more insight (45 minutes) | Back to top

Watch a video (19 minutes) | Back to top

We draw a more concrete connection between evolutionary game theory population dynamics and business transaction payoff matrix analysis. In this video, population dynamics equations are derived in a way so that it seems as though cells repeatedly face the consequences of games each time they participate in cell-cell collisions.

In the videos so far, we can identify two propositions.

  1. Proposition 1: What happens to an agent depends, not only on the agent's strategy, but also on the strategies presented by other agents.
  2. Proposition 2: An agent chooses to adopt a strategy by thinking about and weighing the consequences it might face in hypothetical scenarios in which its strategies and the strategies of other agents are combined in different ways.

How can the quantitative analyses we have explored help us to relate these propositions?

Watch a video (11 minutes) | Back to top

In this video, we provide an example how the concept of time can connect proposition 1 to proposition 2.

Glance at literature | Back to top

Please flip through figures from these papers.

We have so far focused on well-mixed models. What happens when populations are not-as-well-mixed or not mixed at all? To address this question, work through the next section following the break.

Break (5 minutes)

“

. . . Volterra's equations for the dynamics of a predator and prey species . . . [in] a sense . . . are manifestly false. . . . . Their merit is to show that even the simplest possible model of such an interaction leads to sustained oscillation -- a conclusion it would have been hard to reach by pure verbal reasoning.

--John Maynard Smith (1982) Evolution and the Theory of Games, Cambridge University Press, p. 9

Spatiality (40 minutes) | Back to top

Read through this multiple-choice prompt with a partner: Two populations of annual plants, “cooperators” and “defectors,” are sown on a field. Prisoner’s dilemmas describe the chemical and mechanical contact that occurs repeatedly between pairs of neighboring plants. These interactions determine the numbers of seeds that the plants contribute to the next generation. Which statement is true?
  1. Increasing the spatial area over which offspring randomly disperse spreads the offspring of defectors too thin, making it more difficult for defectors to compete with dense pockets of cooperators. These pockets of cooperators survive and perpetuate heterogeneous co-existence.
  2. Increasing the spatial area over which offspring randomly disperse promotes heterogeneous co-existence because survival of cooperators relies on their ability to move away from defectors in an ongoing cat-and-mouse chase.
  3. Increasing the spatial area over which offspring randomly disperse makes it easier for defectors to take over the lattice and more difficult to realize heterogeneous co-existence.
  4. Increasing the spatial area over which offspring randomly disperse promotes heterogeneous co-existence because defectors and cooperators both have increased chances of bumping into other cooperators.
  5. None of the above
Watch a video (4 minutes) | Back to top

One of the basic lessons in spatially-resolved models is that social interactions can generate different outcomes depending on whether dispersal is localized or global.

Work with a partner to address the multiple-choice prompt form the beginning of this section.

Glance at literature: We used the prisoner's dilemma in the preceding video because it is compact and simple. To see that the utility of the familiarity we have developed extends beyond this particular game, consider the game of rock-paper-scissors. In rock-paper-scissors, there are three, rather than two cell types. Additionally, there is no obvious hierarchy (unlike the prisoner's dilemma, in which defectors tend to outperform cooperators when mixed). Nonetheless, keeping mixture local (rather than global) also promotes coexistence in this system, as experimentally and theoretically explored in Kerr B, Riley MA, Feldman MW, and Bohannan JM (2002) Local dispersal promotes biodiversity in a real-life game of rock-paper-scissors. Nature 418:171--174. Please see figures 2 and 3. The manuscript is made available at the first author's research webpage.

Interactive simulation (15 minutes) | Back to top

To play with a rock-paper-scissors style cellular automata simulation, click on the "Start MJCell" button to the right. A Java applet will launch in a separate window (example screen capture below). This applet is Mirek's Java Cellebration. Mirek Wójtowicz has made the source code available for modification. See also the list of cellular automata Java applets assembled by David Griffeath (Mathematics, University of Wisconsin).

Three elements are highlighted in the example screen capture of the applet. The applet will load rule 313 from the Cyclic CA family. This is represented near the question-mark button (?) using the notation R1/T3/C3/NM.

There are three cell types (C = 3), which we can call rock, paper, or scissors. Each cell type can be replaced by a cell of the next type (rock defeats scissors, scissors defeat paper, and paper defeats rock). A cell surveys its neighbors. In this simple example, the neighbors are the 8 nearest sites (N, W, S, E, NW, SW, SE, NE) that a cell can get to by moving at most one lattice bond (horizontal, vertical, and/or diagonal). This is notated by saying the range (R = 1) is one bond (see the entry on Moore neighborhood from Wolfram MathWorld). If there are at least a threshold number of cells (here, T = 3) of "the next" cell type among the neighbors, the cell switches. For example, if rock is surrounded by 3 papers, rock becomes paper. The R/T/C/N notation is described in detail in the lexicon for the applet.

Work with a partner to address the following prompts.
  1. Change the seeding percentage from 20% to 60% and press Rand to reseed the lattice. Press START. What patterns do you see?
  2. As we said, the current rule is R1/T3/C3/NM. This means that each cell is looking among its 8 neighbors to see whether at least 3 (37.5%) of them are of the next cell type. Consider a larger neighborhood, for example, with R = 7. How many neighbors are there? What threshold (T) is 37.5% of the neighbors?
  3. Press the question mark button to change the rule to R7/T??/C3/NM, where ?? is determined in the previous prompt. Reseed the lattice at 60%. Press START. Now what do you see? Try reseeding the lattice a couple times.

Are you a dork? If so, you might be interested in rock-paper-scissors-lizard-spock.

In this section, we established intuition that spatial isolation can promote heterogeneous coexistence while thorough mixture can promote homogeneity. The kind of interaction (e.g. seed dispersal and surveying of neighbors) we have discussed is, in a sense, context independent. It occurs to the same spatial extent regardless of whether the environment in which cells/plants find themselves. Life is not always so simple. An example of conditional movement occurs when parents base their decisions to leave town because they don't like the neighbors. How does conditional movement help us to refine the intuitions we have developed using the previous video? The next section provides a way to address this point.

Conditional movement (10 minutes) | Back to top

Spend 10 minutes working through the figures on Athena Aktipis's "walk away" description. Work with a partner to practice presenting the figures in plain language.

In addition to movement, other phenotypes can be contextually triggered. For examples in ant colonies, watch the video in the next section.

Conditional task switching (25 minutes) | Back to top

In this microworkshop, we have been informal regarding the word "strategy" and "behavior." The purpose of this prompt is to encourage increased precision. Use the following prompt to start thinking about differences between "strategies" and "actions" or "behaviors." Without spending too much time on responding to the prompt, go ahead and watch the video. While Deborah Gordon's talk is not meant to address these linguistic concerns directly, the video will provide 20 minutes of distraction for your brain to "solve the problem without really directly working on the problem." After watching the video, work on the prompt for no more than about 10-15 minutes. Finally, consolidate your understanding by looking up the definition to a "complete contingency plan."

Read this prompt with a partner: Sometimes, an ant can give up a task and take on another. The fraction of time spent on a task depends on the demographic composition that the ant infers from recent ant-ant contact. How many strategies are there in each of the following situations?

  1. Each ant could be at any moment performing any one of tasks A, B, and C. Task switching occurs between each pair of tasks.
  2. Again, the three tasks that can be performed are A, B, and C. An ant can switch from task A to task B and vice versa. An ant that performs task C does not perform any other task.

Watch a video (21 minutes) | Back to top

Deborah Gordon (entomologist from Stanford) talks about task allocation and organization without central control in ant colonies.

Work with a partner to address the prompt at the top of this section.

Pencil-and-paper activity: Evolutionary game theory for biologists | Back to top

The following files are from the dinner talk on Monday, 2013 August 12 given at the Princeton/Johns Hopkins Game Theory and Cancer Workshop in Baltimore, MD. This presentation has not yet been rendered into an audio-narrated video. To request to see the presentation by Skype conference, please send an email (priority for Physical Sciences Oncology Network members and patient advocates).

Creative Commons License © Copyright 2011-2015 David Liao. These videos and slides are open course ware made available under a Creative Commons license (CC BY-SA 4.0). The lightbox and social sharing effects are scripts by Stéphane Caron (CC BY 2.5).