Game theory, which provides tools for studying strategic interactions between rational economic actors, can offer useful insights into situations with network externalities. In particular, real life situations involving complicated interactions between individuals can be analyzed through simple games designed to capture the salient features of the situation. Game theory can then help to figure out why some outcomes emerge and why others do not. We can apply game theory to the new dollar coin problem to help us understand the nature of the network externalities that arise when a new coin is introduced.
Consider the following game between the public and businesses. The public must decide whether to use the new onedollar coins or continue to use one dollar notes, whereas businesses must decide whether to upgrade their vending machines to accept onedollar coins instead of notes. We assign a payoff to the different outcomes, for each economic actor. (A) If the public uses notes and businesses do not upgrade their machines (the status quo) we assign a payoff equal to 1 to both the public and businesses. (B) If the public uses the coin and businesses upgrade their machines (the efficient outcome) the payoffs of both the public and businesses equal 2. (C) If the public uses notes but businesses upgrade their machines, the public’s payoff becomes 0, assuming buyers cannot use their notes in the upgraded vending machines, and businesses’ payoff equals 1 since they bear the upgrading cost. This is the worst outcome for businesses. (D) Finally, if the public uses the coin but businesses do not upgrade their machines, businesses’ payoff becomes 0 since the vending machines cannot accept buyers’ money, and buyers’ payoff equals 1 since they bear a cost to exchange notes for coins. This is the worst outcome for the public. The game and the assumed payoffs are summarized in the table below, where the two columns represent businesses’ choices and the two rows indicate the possible choices of the public. The first number in each cell is the public’s payoff, and the second number is the businesses’ payoff.

Upgrade 
Do not upgrade 
Uses coins 
2, 2 (B) 
1, 0 (D) 
Uses notes 
0, 1 (C) 
1, 1 (A) 
Consider the situation in which the public uses coins and businesses upgrade their vending machines (B). Since both economic actors get their highest payoffs, neither the public nor businesses have any reason to choose another action. Therefore, game theory suggests that this outcome can occur: in the language of game theory, this is an equilibrium outcome. Now, consider the situation in which the public sticks to the notes and businesses do not upgrade their vending machines (A). The public has no reason to use coins (given that businesses do not upgrade) since the payoff it would get for using coins would be 1, whereas the payoff it gets for using notes is 1. Similarly, businesses have no incentive to upgrade their machines (given that the public uses notes) as it would lower their payoff from 1 to 1. Therefore, this situation is also an equilibrium. However, this equilibrium is inefficient since it would be beneficial for all to switch to the new coin and to upgrade the vending machines. Finally, there is a third equilibrium situation in which half of the public adopts the new coin and half of the businesses upgrade their vending machines. Although game theory suggests that the worst outcome (C and D above) will not prevail, the most efficient outcome (B above) may also not win out. 