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  "body": "\n## Summary\n\nThe Ehrenfeucht–Fraïssé game is a two-player game played on two [[structures|983GZBQYT4]] of the same signature.\nOne player, Spoiler, tries to show that the structures differ, and the other, Duplicator, tries to show that they look alike.\nFor a finite relational signature, Duplicator wins the $n$-round game exactly when the structures satisfy the same sentences of quantifier depth at most $n$.\nThe game is a main tool to prove that a property is not first-order definable, in particular in finite model theory.\n\n## Explanations\n\n### L1 human\n\nIn each round, Spoiler points at an element of one of the two structures, and Duplicator answers with an element of the other.\nAfter all rounds, Duplicator wins if the chosen elements relate to each other in the same way on both sides.\nOn the linear orders with $3$ and $4$ elements, Duplicator wins the game with $2$ rounds.\nSpoiler wins with $3$ rounds, so some first-order sentence with three nested quantifiers tells the two orders apart.\n",
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