Price of Anarchy, explained.
The price of anarchy is the ratio between the cost of a worst equilibrium and the lowest feasible system cost, under a specified game and cost measure. It quantifies a possible efficiency loss from individual incentives.
Why it happens
Each infinitesimal driver chooses a route with minimum personal delay. At 60 units on the variable road, both roads take 60 minutes and no driver improves by switching. But minimising total delay also accounts for the delay each extra user imposes on everyone already on that road.
The price of anarchy compares the cost of a worst equilibrium with the minimum achievable system cost under a stated model. Individual incentives can produce an inefficient collective outcome.
Read the result
Compare Selfish equilibrium with System optimum, then try intermediate slider allocations. The displayed 1.18 ratio remains the equilibrium-to-optimum comparison; your chosen allocation has its own total cost. At the optimum, some users still prefer a faster route, so individual incentives alone do not maintain it.
A worked example
Two roads carrying one hundred traffic units
At the equilibrium, 60 use the variable road and 40 use the fixed road. Both take 60 minutes, giving 6,000 unit-minutes in total.
Route 30 to the variable road and 70 to the fixed road: 30×30+70×60=5,100 unit-minutes.
The ratio 6,000/5,100 is about 1.18. The example shows inefficiency in these invented roads, not a universal bound for traffic networks.
OPTIONAL DEEPER DETAILGo deeper: inside the model
Inside this model
The variable road takes x minutes when carrying x traffic units. The other road always takes 60 minutes. Total delay is x² + 60(100−x). The nonatomic equilibrium has x=60; the system optimum has x=30. Their ratio is 6000/5100, about 1.18. The slider explores allocations; it does not change that equilibrium ratio.
Where this idea is useful
A practical use
Routing rules can improve total travel time while asking some users to choose a route that is not their individual minimum at that allocation.
A common misconception
“An equilibrium is automatically the best collective outcome.”
Equilibrium describes incentives to deviate, while optimality describes a chosen system objective. External congestion costs can separate those two ideas.
What this explanation leaves out
- Traffic is divisible and users are infinitesimal. These are arbitrary delay units, not a city forecast. The number shown is specific to these two roads, not the general worst-case bound.
Why would fixed-road users resist the system optimum?
At that allocation their road takes 60 minutes while the variable road takes 30. An individual infinitesimal user benefits by switching, even though many such switches raise the total delay.
Which individual shortcut in your system imposes a cost on others that its user does not consider?
Associated thinkers
Associations marked provisional are awaiting source review.
Further reading
Explore the original research or the teaching reference behind this experiment.