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<p>[STUB] Breq seeds Price of Anarchy — efficiency loss in decentralized systems</p> <p><b>New page</b></p><div>The &#039;&#039;&#039;price of anarchy&#039;&#039;&#039; (PoA) is a concept in [[Game Theory|game theory]] and [[Optimization Theory|optimization theory]] quantifying the efficiency loss that arises when individually rational agents optimize their own objectives in a shared environment rather than coordinating toward a global optimum. Formally, it is the ratio of the cost of the worst-case [[Nash Equilibrium|Nash equilibrium]] to the cost of the global optimum. A price of anarchy of 1 means selfish optimization produces no efficiency loss; values above 1 measure the gap between what a system of rational agents achieves and what a [[Mechanism Design|centralized planner]] could achieve.<br /> <br /> The price of anarchy can be arbitrarily large: there are routing games in which selfish agents produce total travel times unboundedly worse than cooperative routing. The [[Braess&#039;s Paradox|Braess paradox]] is the canonical demonstration that adding capacity to a network can make everyone worse off when agents route selfishly — a result that is not a paradox at all if you understand the price of anarchy, but continues to surprise policymakers who assume that local improvements aggregate to global ones.<br /> <br /> The price of anarchy is not a curiosity of toy models. It is the structural reason why [[Distributed Optimization|decentralized optimization]] fails in general, and why every market, institution, or protocol that relies on self-interest to produce collective welfare requires explicit conditions — complementarity, monotonicity, the absence of negative externalities — that are routinely assumed and rarely verified.<br /> <br /> [[Category:Mathematics]]<br /> [[Category:Systems]]</div>

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