How to use Game Theory and Decision Trees

In 2018, a news story came out that Apple was going to investing $319 million dollars in a partnership with an augmented reality hardware supplier (Collins, 2019).

If you were Apple, how would you decide this idea was a good one? How much would you be willing to spend? How would you respond if you were Samsung?

We can analyze answers to these questions using game theory and decision trees.

These flexible tools can be used for any decisions that involve other parties and probabilistic scenarios; something that product managers deal with all the time.

Caveat: For this example, I have made assumptions on some of the numbers involved. When you apply this methodology for real, you should invest time in getting the correct numbers if you can. Even if you can’t get all the numbers, this framework is still valuable. It can highlight the breakeven point on a given number or the value of getting that information.

Apple’s Situation

Augmented Reality technology was a trending technology Apple was looking to invest in. To support this in their phones, Apple required a complicated laser component called a VC cell. This is used for face mapping and other augmented reality applications.

VC cells were currently manufactured by two parties:

  • Lument0 – existing Apple supplier
  • Finisar – existing supplier to both Apple & Samsung

Demand was outstripping supply by quite a margin. Meaning Apple couldn’t get as many of these parts as they wanted.

Now that we understand the situation, we can follow the steps to create a Decision Tree.

Step 1 – Map out the initial moves

  • Share supply with Samsung (aka “do nothing, and follow status quo”), OR
  • Build additional capacity

Step 2 – List all the possible counter-moves at each branch

If Apply chooses to build, Samsung can respond as such:

  • Share supply with Apple (aka “do nothing and follow status quo”), OR
  • Invest in their own additional capacity

For this analysis we assume Samsung will not build capacity unless Apple does.

Step 3 – List the possible scenarios at each branch

Decisions in the present are often made without certainty in the future. Decision trees can cater for this, by listing possible scenarios and assigning probabilities. These act as weightings to the outcomes as we will see.

Regardless of the moves and counter-moves Apple & Samsung make, there are two possible scenarios in future:

  • That future demand for Augmented Reality phones (and thus for this part) goes up, OR
  • That future demand for these parts goes down

For this example, I have assigned the following probabilities to the scenarios:

  • Probability 80% for high demand
  • Probability 20% for low demand

Step 4 – Assign the payoff values of each termination node

Work through the termination nodes and assign a value to each player. What would their gains be, assuming the game played out this way?

For this game, the payoff rules are as follows:

  1. Apple share capacity + high demand = make equal revenue
  2. Apple share capacity + low demand = make equal revenue, but less than in #1
  3. Apple build capacity + Samsung does nothing + high demand = Apple makes most revenue
  4. Apple build capacity + Samsung does nothing + low demand = make equal revenue, same as #2
  5. Apple build capacity + Samsung builds capacity + high demand = both make high revenue
  6. Apple build capacity + Samsung builds capacity + low demand = make less revenue, same as #2

Did you spot something wrong in the above?

In node #4, where you have more capacity than your rival, but the demand is low (so you can’t utilize it), I’ve said Apple would make the same money as Samsung would. Some might say that Apple would make less, given they’ve effectively wasted investment. However in micro-economics, this is considered a sunk cost and not factored into the payoff value.

That extra capacity may have cost you $1M, it may have cost $100 or it may have cost $1. It doesn’t matter; at any given decision node, you only make the decision based on what you get down each branch; not what you paid to get there.

The investment to build capacity will be factored into our decision – it’s just not represented at the payoff node.

Step 5 – Use backwards induction to predict the moves each party will make

Now that we have the payoffs for each party at each termination node, we can calculate the weighted payoffs, and then determine which decision each party would make, should they find themselves there.

Again, the key is to select the best option open to you, regardless of other potential branches or costs to get there.

To calculate the payoff of a given branch, where multiple probabilistic events occur, apply the formula:

[ (Probability A * Payoff A) + (Probability B + Payoff B) + etc. + etc. ]

Backwards induction means you start at the termination node and work backwards. For each decision, work out the payoffs for that player against each branch. They would then select the optimum branch at that point.

Working backwards from Scenario 1 & 2, the first decision we come to is Apple’s original choice to ‘share or build’. Thus we evaluate the value of ‘sharing’.

  • Scenario 1 ($1B) * .6 + Scenario 2 ($800M) * .4 = $920M

Thus the weighted average payoff for sharing capacity for both parties is $920M.


Evaluating the ‘Build’ branch’:

  • Starting at the Scenario 3 & 4 and working backwards, we can calculate the weighted payoff to be:
    • Apple: $2B * .6 + $800M * .2 = $2.56B
    • Samsung: $800M

Thus Samsung’s payoff to take no action, given they are at this point in the game, is $800M.

Note: as Party B, you do not take into account what Party A will get for a given move. Your sole focus is maximising your return, not minimising theirs.

If Samsung were facing this choice, they would rather take the $1.16B payoff – if it were free. However, building capacity costs money. This is where the investment amount – that we previously discarded as a sunk cost – comes into play.

How much could they afford invest?

The investment threshold is the difference in payoffs. Thus they could spend up to $360M ($1.16B – $800). In hindsight we know that Apple spent $390M. If we take this at face value and assume the cost would be the same for Samsung, building would only return a payoff of $1.16 – $390M = $770M. Thus they would choose not to build, and we can cross out the other branch as illogical.


Based on the above, we know that Samsung won’t build, which gives Apple the potential payoff of $1.52B vs. $920M for sharing.

They could thus invest anything up to or less than the difference in investment. It turns out they come out well ahead, spending only $390M. This means that building capacity should give them a net benefit of $210M.

Summary

Decision trees can be used to get another lens on complex decision making, with multiple parties and probabilistic outcomes.

They can highlight the value of information and identify breakeven numbers and thresholds.

In the example above, Samsung could spend money on market research to check some of the assumptions in the termination nodes. If the probability of this future demand was not certain, or the market value of the high demand was in question, they could use payoff trees to quantify how much to pay to get better data.

As a Product Manager, it’s unlikely you’ll be making decisions on this level. You can however apply the same to decision making on competitive features, getting a feel for how much to invest, how your competition might react, and the payoffs for doing so.

References

Collins, K. (2019). Apple aims to ‘push boundaries’ via iPhone laser chip maker. [online] CNET. Available at: https://www.cnet.com/news/apple-invests-390m-in-finisar-iphone-laser-chip-maker/ [Accessed 24 Mar. 2019].