Experiment 14 Evolutionary Games
Survival of the Strategy
Release a small invading group and watch Hawk and Dove behaviors spread or shrink over generations.
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Start with “Work It Out, Step by Step” below. The optional expert section explains its symbols as you go. For more examples, use the plain-language math guide.
1. Make a Prediction
Who Does Well Among These Neighbors?
Two behaviors compete for a resource. Hawk insists on taking it. Dove yields to a Hawk but shares with another Dove. Two Hawks fight, so conflict can erase the value they gain. The bird names label behaviors; this is not a model of actual birds or a claim about how people should behave.
2. Make Your Choice
3. Reveal the Incentives
An Easy Win Can Become an Expensive Habit
An expected score is the average score over many randomly matched encounters. During selection, when both behaviors are present, the behavior with the higher expected score gains a larger share. Later switching can change the final result. A generation is one update of those shares, not a fixed number of years.
| Your behavior | Meet a Hawk | Meet a Dove |
|---|---|---|
| Hawk | Subtract conflict cost from resource value, then take half | Whole resource value |
| Dove | 0 | Half the resource value |
For Hawk against Hawk, subtract the conflict cost from the resource value first, then divide by 2. With value 4 and cost 8, each Hawk averages a loss of 2.
| Generation | Hawk share | Hawk score | Dove score | Population average |
|---|
Try introducing a Hawk into a mostly Dove population. Then try introducing another Hawk when Hawks already dominate. Winning one encounter and spreading through a population are different questions.
4. Change One Assumption
Change the Resource or the Cost of Fighting
Changing a setting restarts the population. Switching is called mutation in this model: a fixed fraction of each behavior changes to the other after selection. There are only two behaviors, no family histories, and no random birth or death events. Runs stop at 200 generations.
Work It Out, Step by Step
Use a resource worth 4 points, conflict costing 8, and a population that is half Hawk. Leave switching at 0% for this example.
- Price each kind of meeting. Two Hawks each average \((4-8)/2=-2\). A Hawk meeting a Dove gets 4. A Dove meeting a Hawk gets 0. Two Doves get 2 each.
- Find the Hawk average. Half the encounters lose 2 points and half earn 4 points. Add half of each: \((-2)/2+4/2=-1+2=1\).
- Find the Dove average. Half earn 0 and half earn 2: \(0/2+2/2=1\). Both behaviors average 1 point, so neither grows through selection at this mixture.
- Start with fewer Hawks. With only 20% Hawks, a Hawk meets a Hawk about 20 times in 100 and a Dove about 80 times. Its average is \(0.2\times(-2)+0.8\times4=2.8\). A Dove averages \(0.2\times0+0.8\times2=1.6\). Hawks now spread.
- Let success change the surroundings. As Hawks spread, they meet each other more often and pay more conflict costs. For these settings, a mixture near half Hawk and half Dove balances their scores. Without switching, a population containing only one behavior stays that way until you introduce the other.
Selection means the higher-scoring behavior becomes more common through the update rule. Invasion means introducing a small amount of a behavior and asking whether its share then grows. Neither term says that the resulting population earns the highest possible combined score.
For experts: formal model and assumptions
Expected Scores in a Randomly Matched Population
- \(p\), \(V\), and \(C\)In words: “Hawk fraction,” “resource value,” and “conflict cost.”
- A fraction p of 0.2 means 20% Hawk. Then \(1-p\), read “one minus p,” is the Dove fraction. V and C are nonnegative point amounts.
- \(f_H\), \(f_D\), and \(\bar f\)In words: “Hawk’s average score,” “Dove’s average score,” and “the population’s average score.”
- The small H and D are labels called subscripts. The bar over f marks the average across the population. An expected score weights each possible score by how often that encounter occurs.
In words: “For a Hawk, weight the fighting score by the Hawk fraction and the full resource by the Dove fraction. For a Dove, only a Dove encounter pays. Then weight both behavior scores by their shares to get the population average.” Adjacent letters and parentheses mean multiply.
The Discrete Update Actually Used
Discrete means the app moves in separate generation steps. It uses a proportional reproduction rule, also called a discrete replicator update, followed by symmetric switching. “Symmetric” means the switching probability is the same in both directions.
- \(B\), \(q\), and \(p_{\mathrm{next}}\)In words: “positive baseline,” “Hawk fraction after selection,” and “Hawk fraction next generation.”
- B is an amount added to every score to keep reproduction weights positive, even when encounter scores are negative. q is the intermediate fraction before switching. The lower word “next” is a label, not multiplication.
- \(\mu\)In words: “mu,” the switching fraction.
- The Greek letter mu names the chance of changing to the other behavior. A 2% setting means \(\mu=0.02\). See rounds and updates.
In words: “Add one, resource value, and conflict cost to set the baseline. Multiply the Hawk share by its positive reproduction weight, then divide by the population’s average weight. Afterward, some Hawks switch away and some Doves switch in.” The last formula is the shortened form of keeping \((1-\mu)q\) Hawks and adding \(\mu(1-q)\) Doves who switch.
The baseline determines the speed of selection; it does not change which behavior has the higher encounter score. This is an infinite-population, random-matching, deterministic approximation. “Infinite population” means fractions change smoothly without rounding to whole individuals. “Deterministic” means the same inputs always produce the same path. A release replaces 2% of the current mix with the chosen behavior and is not an additional ongoing mutation rate.
With switching off and \(0<V<C\), equal scores occur at \(p=V/C\). In words: “If resource value is positive and less than conflict cost, the balancing Hawk fraction equals value divided by cost.” When \(V\geq C\), Hawk has at least as high a score throughout the population range. Here \(<\) means less than and \(\geq\) means greater than or equal to. The no-switching interior balance is a useful reference; switching generally changes it.
Evolutionary stability asks whether a resident behavior or mixture resists sufficiently small alternative invasions under a specified model. A single 2% release over 20 generations is an illustration, not a proof covering every possible invading strategy. This lab includes only Hawk and Dove and does not track individual family histories.
Concept reference: MIT’s lecture on evolution, learning, and the Hawk–Dove game.
5. Take It Somewhere Else
What If Everyone Copies the Shortcut?
A team gains an advantage by taking a scarce shared resource before anyone else. Other teams copy that behavior. Would the original advantage necessarily survive?
Reveal the reasoning
No. Copying changes the environment. A tactic that works when rare can become costly when everyone uses it. Check the frequency of the behavior, the cost of conflicts, and the rule that makes it spread. The most common behavior need not produce the best group outcome.
A Model Is a Place to Start
These small models make the incentives visible. Their results follow from their stated rules; they are not forecasts of how every person or organization behaves. A simulated strategy is a rule, not a personality.
Scenario links save the controls and random seed. To reproduce an interactive run, make the same choices in the same order. Changing a setting restarts the experiment.