Game Theory

Game Theory Parables for Game Designers

Another Cooperative Dilemma: Action Selection

Consider an action selection game where an action only occurs if both players select it.  If the two players miss-coordinate -- if they select different actions -- then the players get nothing.  For the sake of discussion, suppose there are only two actions, Develop and Export, and that players select their action simultaneously.  If they both select Develop, that action happens, and the same goes for Export.  If they mis-match, however, nothing happens and the players get no benefit that round.

To make this example more interesting, suppose that one player prefers the Develop action -- suppose they get more points, or advance their strategy further, or build their engine faster -- while the other prefers the Export action.  Both players prefer that something happens instead of nothing.

As a strategic matrix, here is the Action Selection game:

DE 1.jpg

The row player prefers the Develop outcome, while the column player prefers the Export outcome, and neither of them puts any utility on the outcome where nothing happens.

If the action selection is to be simultaneous and independent, what should the players do?

This is not a situation like the Prisoner's Dilemma where there is only one option -- no strategy dominates the other for either player.  Nor is it quite like the Stag Hunt, where one strategy at least promised a minimal, risk-free payoff.  It does, however, have the same sort of coordination issue that the Stag Hunt has, as players do want to coordinate, but on which outcome?

Each player has their preference for the chosen outcome, but since those are different, there is no natural agreement.  In a longer game where this choice is repeated, players could consider taking turns: Develop one turn, Export the next.  There are always issues with finitely repeated games (to be discussed elsewhere), especially if the game is competitive in nature.

Another option available to Game Designers is to rely on a Correlated Equilibrium, which is a lot more straightforward than it sounds.  If the game had a randomizing element in it, even a simple coin flip, the players could use that to coordinate their actions.  If they decide to play Develop on Heads, for example, and Export on Tails, they actually arrive at a stable solution to this game.

Players have no incentive to unilaterally deviate from the suggestions of the coin flip.  If it comes up Heads, each player knows that the other will pick Develop.  If the Export-preferring player thinks of switching to Export, they know they will get 0 utility instead of 4.  It is in their interests to stick with the result of the coin flip.  This means that players have found a way to coordinate by using only a randomizing device.

Sacrificing Points

There is another way for one player to force the issue and get the outcome they prefer, and it too involves an unorthodox addition to the game.  Imagine if one player was given the opportunity to sacrifice 2 victory points before choosing their action.  This gives the row player four strategies for the bigger game:

      • Sacrifice points, then play Develop

      • Sacrifice points, then play Export

      • Do not sacrifice, then play Develop

      • Do not sacrifice, then play Export

In order to have a complete set of strategies for this bigger game, the column player must specify what they will pick if the row player does, or does not, sacrifice points.  That is, the column player's strategies are a pair <x, y> where 'x' is what they will play if the row player sacrifices points, and 'y' is what they will play if no points are sacrificed.  Row player's strategies are:

      • <Demand, Demand>

      • <Demand, Export>

      • <Export, Demand>

      • <Export, Export>

Each player has four strategies in this larger game, resulting in a 4x4 matrix instead of the 2x2 matrix of the original game.  Skipping the details of the solution (which uses a process called Iterated Elimination of Weakly Dominated Strategies), there is one unique outcome: row player will not sacrifice points and will play Develop, and column player will play Develop.  

This is an interesting outcome because all that needs to happen is that the row player is given the opportunity to sacrifice points.  They never actually do it, but the possibility of it is enough to secure the Develop action.  Giving that player the opportunity to do something irrational (and hence the opportunity for them to not do it) is enough to get both players to coordinate on row player's chosen action.

One interpretation of this curious fact is that row player, by choosing not to sacrifice points, is signalling that they expecting an outcome that is better than any possible outcome that results from sacrificing points in the first place.  The only outcome that meets that criteria is that of mutual Develop.  The column player expects the row player to behave in a certain way, so must strategically react to this and also pick Develop.

Here is an example where the ability to sacrifice points can lead the players to mutually beneficial cooperation.  Consider this simple example, where each player picks a railroad to invest in.  Imagine that investments yield payoffs according to the following matrix:

BO1.jpg

The row player can invest in Baltimore or Ohio, while the column player can invest in Northwestern or Southern.  By identifying best responses, this game can be seen to have one Nash Equilibirum -- one outcome where neither of the players can increase their payoff by unilaterally changing their strategy.  That equilibrium occurs where row player invests in Baltimore, and the column player in Northwestern, yielding 1 point for row player and 3 for column.

Notice that the outcome from Ohio + Southern is strictly better for both players (giving 3 points for row player and 4 for column), but it is not a stable equilibrium.  If row player knows that column will invest in Southern, they will switch from Ohio to Baltimore, gaining 4 points instead of 3.  Likewise, if column player knows that row will actually invest in Baltimore, they will switch to Northwestern, gaining 3 points instead of 1.

There is an unorthodox way to get both players to end up at the strictly better outcome of Ohio + Southern: give row player the ability to enter into a binding deal that forces a 2 point penalty if they invest in Baltimore.  Why would row player ever agree to a deal that can only impose a penalty?  Because it turns the game into this:

BO2.jpg

This game also has one Nash Equilibrium, and it is where row player invests in Ohio and column player invests in Southern.  By giving the row player the ability to enter a binding contract which serves only to punish the choice of one action, both players end up better off than they were before, in the previous version of the game.  What is happening is that, by accepting the contract, the row player is signalling to the column player that the choice of Baltimore is not a Strongly Dominant strategy, that there is a situation where that player would prefer to play Ohio instead.  

Once again, as in the Action Selection example from earlier, the row player doesn't actually sacrifice the points (as they do not select Baltimore), but the existence of that penalty is enough to choose a different action.  The column player knows this, which results in a better outcome for all.

Sam Hillier