Coordination Games



Coordination game

Coordination is required in every aspect of day to day activities.  It can be coordination between husband and wife, for research activities, in politics and economics, or it can be for a choice of a computer technologies. All the players advantage by coordination.  For example, a situation of choosing sides on the road while driving. Consider the encounter of two vehicles on a narrow dirt road. Both drivers will successfully pass each other if they both choose different sides on the road; failure to do so will lead to an accident. Figure 1, shows a payoff matrix for such a game wherein each player receives a payoff of 5 for choosing the same side. Therefore, coordination is required for successfully crossing each other, and hence a coordination game is being played between the players.

Player 2
Player 1 Right Left
Left 5,5 0,0
Right 0,0 5,5

Figure 1: Choosing Sides

Technically, a coordination game is a game with multiple pure strategy Nash equilibriums, wherein players are better off selecting the same strategies. Payoff matrix in figure 2 represents the general form of 2 ×2 coordination game for which inequalities: a>c; d>b ,and w>y; z>x  hold. Here, {C, C}, and {D, D} are pure Nash equilibriums.

Player 2
Player 1 C D
C a, w b, x
D c, y d, z

Figure 2: Coordination game

Example of Coordination Game

Battle of the sexes is one of the most discussed coordination game. The story is about a couple trying to determine whether to go for a romantic or an action movie. General notion is that man would prefer going to an action movie, while female would prefer going to a romantic movie. Couple would still prefer going to the same movie rather than going to a different movie and spending time alone. Figure 3 shows a payoff matrix for Battle of sexes, where husband choose rows and wife choose columns. There are two pure strategy Nash equilibriums wherein both chooses a same place for going. The game represents an interesting scenario; one of the players consistently performs better than other.

Husband
Wife R A
R 3, 1 0,0
S 0,0 3, 1

Figure 3: Battle of sexes

Stag hunt, a coordination game, symbolizes conflict between individual satisfaction and social cooperation. Two hunters have decision to make, whether to hunt a stag or a hare. Hunting a hare is relatively easy, which makes a tasty but less satisfying meal. On the other hand, hunting a stag is difficult and requires coordination between hunters. Stag makes a large mean and is beneficial for the society on the whole, but requires its member to coordinate.  Figure 4 shows the payoff matrix for the stag hunt game. Again, there are two pure strategy Nash equilibriums in this game. General form of the game, based on figure 2, can be represented by inequalities a > c ≥ d > b and w > x ≥ z > y.

Player 1
Player 2 Stag Hare
Stag 4,4 0, 2
Hare 2, 0 1, 1

Figure 4: Stag Hunt

Finally, a pure coordination game is similar to the game of choosing sides. Again consider a couple trying to decide whether to go out for party or to stay home. Figure 5 shows a payoff matrix for such a game. There are two Nash equilibriums in the game, but both the payers prefer a single equilibrium over other i.e. {Party, Party}.  Such a game where a single equilibrium is preferred by all the players is called pure coordination game. With reference to figure 1, general form of the game is: a>d>b; a>d>c ,and w>z>y; w>z>x

Player 2
Player 1 Party Home
Party 15,15 0,0
Right 0,0 5,5

Figure 5: Pure Coordination game

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