The Emergence of Conventions- The Crossroads Game

Game theory can explain the emergence of road rules, as the outcome of two drivers passing each other depends on their spontaneous actions and not on a pre-designed system of rules.

Consider the following example (after a model by Dr Dieter Schmidtchen (30) and a model by Dr. Bertrand Lemennicier (31).

More on Dr. Schmidtchen

More on Dr. Lemennicier

Suppose that a random driver, Driver 1 (D 1) is approaching an intersection. Another car driving in the opposite direction by Driver 2 (D 2) gets at the intersection at the same time.

Since both drivers have to cross the intersection, D 1 has two options: he can either slow down (strategy S) and let the other car pass or he can maintain speed (strategy M). Suppose D 1 has an expectation of ? that D 2 will slow down and an expectation of 1 - ? that D 2 will maintain speed, where ? ? (0.1) Suppose D 2 has an expectation of ? that D 1 will slow down and an expectation of 1 - ? that D 1 will maintain speed, where ? ? (0.1).

Below are the payoffs measured in the time drivers lose at the intersection. Suppose that if one of the drivers slows down and the other one maintains speed, the one who slowes down will lose one minute and that if both drivers slow down, they will each lose 10 minutes, as they have to decide who will gain priority in crossing the road. Also, suppose that if both drivers maintain speed (and thus collide), they will each lose 21 minutes (assess and agree on the damages, exchange insurance information, etc.), in addition to later on having to incur some cost for repairing their cars.

Description

Assume that the game is played by a very large number of drivers, who have to encounter this situation very often. The players do not know each other’s actions (i.e. the game is not valid if one of the drivers maintains speed as he knows that the other driver will slow down, due to the fact that he has previously encountered him). The only thing known to the players is the game’s payoffs.

The expected loss for D 1 from adopting strategy M is:

EM 1= (1- ?)(21) + (?)(0) , ( D 1's payoff for maintaining speed (21) X D 1’s expectation that D 2 will maintain speed (1- ?) + D 1's payoff for slowing down (0) X D 1’s expectation that D 2 will slow down (?))

The expected loss for D 1 from adopting strategy S is:

ES 1= (1- ?)(1) + (?)(10)

The expected loss for D 2 from adopting strategy M is:

EM 2= (1- ?)(21) + (?)(0)

The expected loss for D 2 from adopting strategy S is:

ES 2= (1- ?)(1) + (?)(10)

Analysis for D 1

The decision on which strategy to adopt depends on each driver’s expectations on what the other driver will do. If, for example, ? = 1, which means that D 1 completely expects D 2 to slow down, D 1 will maintain speed, as his incurred losses will equal 0. If however,  ? = 0, which means that D 1 completely expects D 2 to maintain speed, D 1 will slow down as the losses of him slowing down (1 minute) are smaller than the losses of him maintaining speed (21 minutes).

Supposing that D 1 expects D 2 to have an equal chance of maintaining speed or slowing down,

EM = 21/2 + 0/2 = 10.5

ES = 10/2 + 1/2 = 5.5

Since EM > ES, D 1 will always maintain speed

Driver D 1 will be indifferent between slowing down and maintaining speed when EM = ES , which will happen when the expected loss for D 1 from adopting strategy M equals the expected loss for D 1 from adopting strategy S, which means that EM = ES .

(1- ?)(21) + (?)(0) = (1- ?)(1) + (?)(10) ?= 2/3= 0.67

Thus, when ?< 0.67 D 1 should maintain speed and when ?> 0.67 D 1 should slow down.

Analysis for D 2

Same as analysis for D 1. When ?< 0.67 D 2 should maintain speed and when ?> 0.67 D 2 should slow down.

Nash Equilibria of the Game

There are three combinations (?,?) in which no tendency of change in probabilities arise: (0,1), (1,0) and (0.67, 0.67). While (0,1) and (1,0)are stable equilibria, (0.67, 0.67)is un unstable equilibrium, as seem in the figure below:

Description

Let x1 = probability of D 1 to maintain speed and x2 = probability of D 1 to slow down

Let y1 = probability of D 2 to maintain speed and y2 = probability of D 2 to slow down

Both drivers want to minimize their expected losses

D 1’s expected losses = x1 [(1- ?) (21) + (?)(0)] + x2 [(1- ?) (1) + (?)(10)]

D2’s expected losses= y1 [(1- ?) (21) + (?)(0)] + y2 [(1- ?) (1) + (?)(10)]

A Nash equilibria will consist of probabilities (x1, x2, y1, y2), where x1 = ? and x2= 1- ?

For Driver 1 :

Min(x1, x2) = x1 [(1- ?) (21) + (?)(0)] + x2 [(1- ?) (1) + (?)(10)], where x1 + x2 = 1

The Lagrangian takes the form:

L = x1 [(1- ?) (21) + (?)(0)] + x2 [(1- ?) (1) + (?)(10)] - £ (x1 +x2 -1) - £2x1 - £3x2

When differentiating with respect to x1 and x2 we get the following results:

(1- ?) (21) + (?)(0) = £1 + £2 (1- ?) (1) + (?)(10) = £1 + £3

x1 + x2 = 1

£2 = £3 = 0 which results in

(1- ?) (21) + (?)(0) = £1

(1- ?) (1) + (?)(10) = £1

Thus,

(1- ?) (21) + (?)(0) = (1- ?) (1) + (?)(10),

-20 ? = -30

? = 2/3

Thus,as x1 = 2/3 and x2 = 1/3

x1 [(1- ?)(21) + (?)(0)] + x2 [(1- ?) (1) + (?)(10)] = 2/3 (1-2/3) (21) + 1/3 (1-2/3) = 7

For Driver 2

same method as for Driver 1 leads to

y1 [(1- ?) (21) + (?)(0)] + y2 [(1- ?) (1) + (?)(10)] = 7

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