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February 5

Nash equilibrium when blocking each other the way

A corridor is so narrow that only two people can walk along it side by side. Either you are at the left or you are at the right. If I, being on the right, bump into someone, should I step left? Should the other person step to his right? We prefer to walk straight along the corridor, but our main goal is obviously to go through it. Is here a Nash equilibrium?

Have similar problems to this (with the same metaphors) been treated by math? --Llaanngg (talk) 19:03, 5 February 2017 (UTC)[reply]

I think you are describing the Nash_equilibrium#Coordination_game. -- SGBailey (talk) 19:47, 5 February 2017 (UTC)[reply]

Walking in the middle isn't really an option since neither person gets through, so the choices are walking on the left or on the right. This is the same as the Driving game variation listed in the linked section. According to the article there are actually three Nash equilibria. I think the confusion is from the fact that in a Nash equilibrium each player knows what the other player's strategy is, while in the narrow corridor scenario that's seems unrealistic. For cars, which side of the road you drive on is determined by custom and law, so you can assume what the other player's strategy is and avoid a crash. A similar game occurs when cars meet at a stop sign and one must yield right of way. Drivers are supposed to memorize rules for who is supposed to yield so player's strategies are known and collisions are avoided. But in a corridor it's possible that no such assumption can be made. In that case the consequences of a collision are far less serious though. The strategy would be to pick a side at random and in the 50% of cases you get stuck, pick a new side at random and keep trying until you get through. In real life, players can communicate, even if it's by gestures, to coordinate their moves.

A similar game takes place in the story of Robin Hood and Little John where they meet on a narrow bridge. Each player has the option to yield or stand and fight if necessary. In the story, both players choose to stand and fight, but presumably the payoff matrix is different so it's not clear whether that would be the equilibrium strategy; perhaps yielding would be taken as a sign of cowardice, so it would be preferable to fighting and losing. Also, in that case the fighting abilities of the other player can only be guessed, so the players may have different ideas as to what the actual payoff matrix actually is. --RDBury (talk) 22:02, 5 February 2017 (UTC)[reply]

It isn't quite the "coordination game", but the game theory analysis is easy. There are two Nash equilbria: (1) both players stay to their left, (2) both players stay to their right. In the classic coordination game one of these strategies is superior for both players even though both are Nash equilibria, but in this version both strategies yield identical results. Looie496 (talk) 23:00, 5 February 2017 (UTC)[reply]

There is a third, mixed, equilibrium. -- Meni Rosenfeld (talk) 21:22, 7 February 2017 (UTC)[reply]