Models · Model 006 · A teaching model
The Local Peak
Reform climbs the hill it is on. Search has to go down first.
A society of 200 settles on a peak of a rugged landscape, where height stands for how well a way of living works. Small reforms climb the hill it is on. A higher peak lies across a valley, and every step toward it goes down before it goes up. Turn the levers for search, imitation, the reform story, and a peak that sinks, and watch where the society ends up, and who counts it a peak.
01 · The model
One landscape, your levers
Gold contours are the ground, and height is fitness, one number for how well a way of living works. The pale dots are the society, settled on the old peak, the second-highest on the map. The highest lies across a valley. Press Run, then move the levers. The dashed line on each chart is the same society with reform only and the same luck.
Step 0
A new society on the old peak. Press Run.
The levers
Open dimensions
Mean fitness
This societyReform onlyThe old peak's height
Share in crisis
This societyReform only
Moves, every ten steps
JumpsCopied a neighborWent back to the old peak
02 · The test
300 steps, 9 settings
Each line is 300 steps of one setting, averaged over 20 societies on this landscape. Every society starts on the old peak with the same explorers and the same luck, so the settings alone make the difference. Choose the lines to compare.
20 societies for each setting, 300 steps each.
The numbers
Mean fitness is the average over the 200 agents. An agent is in crisis when its fitness is more than 0.1 below the old peak's height, which is 1.33 on the elite surface. Ended higher is the share of agents standing, at step 300, in a basin whose peak is higher than the old peak was, by 0.05 or more, and a society crossed when half or more of them did. Each number is the average over 20 societies; the lowest point and the worst step are read off the averaged lines above.
| Setting | Mean fitness at the end | Lowest mean | In crisis, worst step | In crisis at the end | Ended higher | Societies that crossed |
|---|---|---|---|---|---|---|
| Reform only | 1.43 | 1.42 at step 1 | 0.1% | 0% | 0% | 0 of 20 |
| Search without imitation | 1.39 | 1.32 at step 60 | 18% | 16% | 4% | 0 of 20 |
| Search with imitation | 1.44 | 1.41 at step 11 | 2% | 0% | 0.9% | 0 of 20 |
| Search and the narrative | 1.44 | 1.42 at step 1 | 0.5% | 0% | 0.6% | 0 of 20 |
| Reform only, the peak sinking | 1.07 | 1.07 at step 300 | 100% | 100% | 0% | 0 of 20 |
| Search with imitation, the peak sinking | 1.87 | 1.34 at step 85 | 99% | 15% | 85% | 17 of 20 |
| Search, imitation, and the narrative, the peak sinking | 1.73 | 1.32 at step 88 | 99% | 30% | 70% | 14 of 20 |
| Search with imitation, the peak sinking, one axis open | 1.07 | 1.07 at step 300 | 100% | 100% | 0% | 0 of 20 |
| Search with imitation, the peak sinking, scored by the popular surface | 1.25 | 1.05 at step 89 | 98% | 5% | 95% | 19 of 20 |
On a peak that holds, search is friction. Explorers alone lower the mean fitness, from 1.43 to 1.39 by step 300, and put about 18 percent of the society in crisis at the worst step, because a long jump in a random direction usually lands on lower ground. Only about 4 percent end in a higher basin. Imitation brings the rest home: the agents who are better off are the ones who never left, so an agent in crisis copies its way back to the old peak, and at most about 2 percent are in crisis at any step. The reform feedback needs no villain, only neighbors who are doing better.
On a sinking peak, reform alone rides it down. The old peak sinks 0.0012 a step, every agent is in crisis by step 85, and none leaves, because every small step away still goes downhill: the mean ends at 1.07. Explorers and imitation are what reach the higher peak, and they reach it through a crisis that touches nearly everyone and then passes. The mean falls to 1.34 at step 85, and by step 300 it stands at 1.87, with 85 percent of agents in a higher basin. In 17 of the 20 societies an explorer lands on the higher peak's slopes and climbs out of crisis there, and imitation carries the rest across. In the other 3 explorers land on those slopes too, but every one of them is in crisis there and copies its way home before it can climb, and the society rides the old peak down like reform alone. The same imitation that saves a society on a holding peak erases the discovery on a sinking one.
The narrative cannot stop the crossing once the peak is sinking, but it thins it. The story recalls agents in crisis to the old peak, and they include explorers who have landed on the far slope and are still climbing. With half of those in crisis recalled each step, 14 of 20 societies cross instead of 17, and 70 percent of agents end higher instead of 85. In the societies that do cross, the crisis passes about as fast as without the story. While the peak holds, the story and the neighbors' example point the same way; once it sinks, the story keeps a few societies home and loses the rest.
Two more rows. With one axis open, the search finds nothing: the higher peak does not lie on the line through the old peak, so the society rides it down like reform alone. Across 20 other landscapes, the same search crosses on 12 with both axes open and on 4 with one. And scored by the popular surface, where the old peak is worth 1.15 and the highest ground is a different hill, the crossing goes the other way: 61 percent end on the popular surface's highest peak, which drew nobody when the elite surface was the measure.
03 · How it works
Ground, steps, jumps, and stories
The ground. Twelve round hills on a square, drawn from a seed, make 7 peaks. Height is fitness, one number for how well a way of living works, which is the biggest thing this toy leaves out, and the note at the end says so again. The ruggedness lever uses fewer, broader hills or more, narrower ones, after Kauffman's NK model, in which more constraints make more peaks. Here more hills make more peaks and higher ground, since the hills add. Lustick brought this vocabulary to institutions, and Russell Weaver and Jason Knight's 2014 article brought it to land use: a society settles on a peak that is not the highest, and the losses it would take to leave are what keep it there.
The old peak. The society starts on the second-highest peak of the elite surface, at 1.43, so a higher one always exists: the highest stands at 2.02, 0.39 of the map away, across a valley. Lustick's phrase for the valley is the right one. A society must become less fit in the process of becoming more fit.
Reform. Each step, each agent tries one move of 0.015 in a random direction and keeps it only if the ground there is higher. That is hill-climbing, the local search in Levinthal's model of adaptation and the exploitation in March's. It finds the top of the hill it is on and never leaves it.
Search. Explorers, a share of the agents, each make one long jump of 0.24 to 0.6 of the map at some step in the first 60 and keep it whatever the ground, Levinthal's long jump. From where they land they climb like everyone else. That is Wright's first phase, a small group drifting off the peak, and March's exploration.
Imitation. An agent in crisis, more than 0.1 below the old peak's height, copies with the imitation probability the spot where a random agent stands, if that agent is better off by 0.05 or more. That is Wright's third phase, a better peak spreading between groups. It is also the engine of the reform feedback: while the peak holds, the agents who are better off are the ones who stayed.
The narrative. With the pull probability, an agent in crisis goes back to the old peak. That is the reform story, come back, it was better. An explorer who goes back before jumping gives up its jump.
Mismatch. The old peak sinks by the mismatch rate each step, in a dip 0.08 wide around it. That is the environment changing under a strategy that does not, the evolutionary mismatch of Russell and Jason's papers and Lustick's Japan after 1990.
Whose fitness. Every hill has an elite height and a popular height, and the slider blends them. The popular surface is the elite surface with three changes: the hills under the old peak count for a fifth less, the hills under the elite's highest peak for two fifths less, and one hill well away from both counts for more, enough to be the highest. Both surfaces are made up. The point is Freire's: the old peak is a peak by somebody's measure, and the slider shows whose. A limit-situation for one group is a slope another group is told is a cliff.
Open dimensions. With one axis open, every move runs along the line through the old peak. A spot can be the top of that line and a slope of the surface, and the higher peak is usually off the line altogether.
The readouts. Mean fitness over the 200 agents. The share in crisis. The share in a higher basin: each agent's spot is climbed to its peak on the ground as it is now, and that peak is compared with the old peak's height plus 0.05. Nothing here is selected but positions. No agent dies, breeds, or is replaced; what changes is where each one stands and whom it copies. Evolutionary language in this model is about configurations and rules, never about people.
- Agents
- 200, on a square map
- Ground
- 12 hills, 7 peaks on the elite surface
- Old peak
- 1.43; the highest is 2.02
- Crisis
- Fitness below 1.33, a tenth under the old peak
- A jump
- 0.24 to 0.6 of the map, once, in the first 60 steps
- Sinking
- 0.0012 a step in the test: from 1.43 to 1.07 in 300 steps
The groundAfter 300 steps of sinkingCrisis below 1.33
04 · What to try
Six experiments
Each button sets up a new society with one setting and starts it running. Every society is different, so run each one a few times. The dashed lines are the same society with reform only.
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Reform only
Everyone takes small uphill steps and keeps the ones that pay. Nobody leaves: the mean fitness holds at 1.43, after the first few steps nobody is in crisis, and nobody ends in a higher basin. Reform cannot see the higher peak, because every step toward it goes downhill first.
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Search without imitation
A fifth of the agents each make one long jump in the first 60 steps. Most land on lower ground. The mean falls to about 1.32 and ends near 1.39, below the old peak, with about 16 percent still in crisis, and about 4 percent of agents reach a higher basin. On a peak that holds, search is friction.
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Search with imitation
Now an agent in crisis copies, one time in five each step, a random agent who is better off. The explorers come home: the mean dips only to about 1.41, at most about 2 percent are in crisis, and about 1 percent end higher. Nobody pulled them back. The neighbors who were better off were the ones on the old peak.
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The narrative
Instead of imitation, the story: half the agents in crisis go back to the old peak each step. On a peak that holds it does imitation's job and does it faster. About half a percent are in crisis at the worst step, and the mean never falls below 1.42. The test shows what the story does once the peak sinks: 14 of 20 societies cross instead of 17.
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The sinking peak
The old peak sinks 0.0012 a step, with a fifth of the agents exploring and imitating at one in five. Reform alone, the dashed line, rides the peak down: everyone is in crisis by step 85 and the mean ends at 1.07. With search and imitation, in 17 of 20 societies an explorer finds the higher peak and the rest follow; in the other 3 the explorers who find it are copied home before they can climb. The mean falls to about 1.34 before it climbs to about 1.87, and 85 percent of agents end higher. The crisis touches nearly everyone, and passes.
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Whose peak
The same sinking peak, scored by the popular surface. On it the old peak is worth 1.15 instead of 1.43, the elite's highest peak 1.21 instead of 2.02, and the highest ground is a hill 0.29 of the map from the old peak on the far side, worth 1.30. The crossing goes the other way: 61 percent of agents end on the popular surface's highest peak, which drew nobody when the elite surface was the measure, and 34 percent on the elite's. Where progress lies depends on whose fitness is counted.
Sources
Reading and notes
- The vocabularyIan S. Lustick, with comments by Daniel Nettle, David Sloan Wilson, Hanna Kokko, and Bradley A. Thayer, Institutional Rigidity and Evolutionary Theory: Trapped on a Local Maximum, Cliodynamics 2(2), 2011. Institutions as hill-climbers stranded on local maxima, and the memos that out-replicate: warnings of immediate pain beat proposals of short-term cost for long-term gain.
- MismatchRussell Weaver and Jason Knight, Evolutionary Mismatch as a General Framework for Land Use Policy and Politics, Land 3(2): 504–523, 2014. The outcome space, the rugged landscape, and the welfare losses from leaving a local maximum that are nonetheless in a trapped society's long-run interest.
- LevelsRussell Weaver, Urban Geography Evolving: Toward an Evolutionary Urban Geography, Quaestiones Geographicae 33(2): 7–18, 2014. Multilevel selection in cities: a peak for the city can be a valley for a block.
- The landscapeSewall Wright, The roles of mutation, inbreeding, crossbreeding and selection in evolution, Proceedings of the Sixth International Congress of Genetics 1: 356–366, 1932. The first landscape, and the shifting balance: drift off a peak, climb another, and let the better peak spread.
- RuggednessStuart A. Kauffman, The Origins of Order: Self-Organization and Selection in Evolution, Oxford University Press, 1993. The NK model: more constraints, more peaks, lower ones.
- SearchDaniel A. Levinthal, Adaptation on Rugged Landscapes, Management Science 43(7): 934–950, 1997. James G. March, Exploration and Exploitation in Organizational Learning, Organization Science 2(1): 71–87, 1991. Local search and long jumps, and why exploitation is effective in the short run and self-destructive in the long run.
- Limit-situationsPaulo Freire, Pedagogy of the Oppressed, translated by Myra Bergman Ramos, Herder and Herder, 1970. Limit-situations and untested feasibility: the slopes that one group is told are cliffs.
This is a teaching toy, not a model of any society. Fitness is one number, and that is its largest simplification: a real landscape has as many heights as there are people on it, and the whose-fitness lever only gestures at that. Its parameters were chosen for teaching, not measured: the hills, the step, the jumps, the imitation, the pull, and the sinking rate, and the seed of the landscape was picked so that the story is easy to see. Both surfaces are made up. Selection here is of positions and rules, never of people: no agent dies, breeds, or is replaced. The code is plain JavaScript and runs in this page.
This model belongs to Russell's work in progress on reform, search, and the polycrisis, one of the strands in his book in progress, Commoning Consciousness.