Game Theory: Incomplete Information
So far, we’ve talked about games where you know everything: all of the player strategies and all of their payoffs. But we are modeling the real world. How often do you know all the options your competition has? How well do you know the costs and benefits of each of their choices? It’s unlikely you know them any better than they know yours, which is not very well.
Most of the time when you are using Game Theory, you might have either incomplete or imperfect knowledge to work from. These sound the same but they are subtly different!
- Incomplete Information means there are things you simply don’t know, such as the opponent’s strategies or payoffs.
- Imperfect Information means you won’t know when or if an opponent makes a move.
When you sit down and model your game, unlike the clean and perfect models I’ve presented so far it’s likely yours will look something like this:
The best way to fill in the gap is to reason through the strategy or payoff based on what you know about the strategy or the player. This can be easier than it sounds as, like in Decision Trees, there are likely a small number of possible outcomes. For example, if you lower the price of your product your competition can only respond in one of three ways: lowering their price, keeping their price or raising their price. Likewise, it might be hard to know the specific payoffs for prices changes to your competition, but you can likely understand the relative payoffs (raising prices produces more revenue while lowering prices reduces it).
If you cannot fill in the gaps then you might want to think about the game differently. So far, the matrix model for games we have been using is called “normal” form. There is another form for game models called “extensive” form, which also captures the order of decisions by players and can help simplifying games with missing information. An extensive form representation of the above example would look as follows if Player 1 goes first:
In extensive form modeling, we start with the first player to make a move (in this case Player 1) and create a graph where the edges are their possible strategies (in this case A or ?, i.e., unknown). Each of those lead to a point where the next player (Player 2) chooses between their strategies which in turn lead to payoffs. This captures the turn-based nature of games where one player makes a move and the other player reacts to it.
As you can see, this greatly simplifies our incomplete knowledge problem because we do not need to worry about all of the unknowns together! We only care about unknowns if the path of decisions that lead to them are likely to occur. For example, if we are Player 2 and Player 1 chooses strategy A, we are going to choose strategy D regardless of what the unknown payoff would be for Player 1 if we had chosen strategy C – because we didn’t choose it!
There are more advanced and formal methods to calculate the unknown payoffs probabilities that we don’t have time to go into today, but in my experience for practical decision making you can get most of the way through simple reasoning and pruning of likely outcomes.
Tomorrow we’ll take everything we’ve covered one step further by covering games with more than two players!
Quote of the Day: “It may be that all games are silly. But then, so are humans. ― Robert Lynd