Introduction

Kaminski et al¹ (Fig 7) provide an example of a decision tree with multiple decision nodes, including some that are descendants of another decision node. This vignette illustrates how rdecision can be used to model a complex decision tree, using the example from Figure 7 of Kamiński et al¹.

The problem

Kaminski et al¹ state the problem as follows:

Consider an investor owning a plot of land, possibly (a priori probability amounting to 70%) hiding shale gas layers. The plot can be sold immediately (800, all prices in $’000). The investor can build a gas extraction unit for a cost of 300. If gas is found, the profit will amount to 2,500 (if not there will be no profit, and no possibility of selling the land). Geological tests can be performed for a cost of 50, and will produce either a positive or negative signal. The sensitivity amounts to 90% and the specificity amounts to 70%. The installation can be built after the test or the land may be sold for 1000 (600) after a positive (negative) test result.

Creating the model

# nodes
d1 <- DecisionNode$new("d1")
d2 <- DecisionNode$new("d2")
d3 <- DecisionNode$new("d3")
c1 <- ChanceNode$new("c1")
c2 <- ChanceNode$new("c2")
c3 <- ChanceNode$new("c3")
c4 <- ChanceNode$new("c4")
t1 <- LeafNode$new("t1")
t2 <- LeafNode$new("t2")
t3 <- LeafNode$new("t3")
t4 <- LeafNode$new("t4")
t5 <- LeafNode$new("t5")
t6 <- LeafNode$new("t6")
t7 <- LeafNode$new("t7")
t8 <- LeafNode$new("t8")
t9 <- LeafNode$new("t9")
# probabilities
p.sens <- 0.9
p.spec <- 0.7
p.gas <- 0.7
p.nogas <- 1.0 - p.gas
p.ptest <- p.sens * p.gas + (1.0 - p.spec) * p.nogas
p.ntest <- (1.0 - p.sens) * p.gas + p.spec * p.nogas
p.gas.ptest <- p.sens * p.gas / p.ptest
p.gas.ntest <- (1.0 - p.sens) * p.gas / p.ntest
# edges
E <- list(
  Action$new(d1, t1, "sell", benefit = 800.0),
  Action$new(d1, c1, "dig", cost = 300.0),
  Reaction$new(c1, t2, p = p.gas, benefit = 2500.0, label = "gas"),
  Reaction$new(c1, t3, p = p.nogas, label = "no gas"),
  Action$new(d1, c2, "test", cost = 50.0),
  Reaction$new(c2, d2, p = p.ntest, label = "negative"),
  Action$new(d2, t4, "sell", benefit = 600.0),
  Action$new(d2, c3, "dig", cost = 300.0),
  Reaction$new(c3, t5, p = p.gas.ntest, benefit = 2500.0, label = "gas"),
  Reaction$new(c3, t6, p = (1.0 - p.gas.ntest), label = "no gas"),
  Reaction$new(c2, d3, p = p.ptest, label = "positive"),
  Action$new(d3, t7, "sell", benefit = 1000.0),
  Action$new(d3, c4, "dig", cost = 300.0),
  Reaction$new(c4, t8, p = p.gas.ptest, benefit = 2500.0, label = "gas"),
  Reaction$new(c4, t9, p = (1.0 - p.gas.ptest), label = "no gas")
)
# tree
V <- list(d1, d2, d3,  c1, c2, c3, c4,  t1, t2, t3, t4, t5, t6, t7, t8, t9)
DT <- DecisionTree$new(V, E)

Tree diagram

Decision tree used in the shale gas problem

Evaluating the strategies

There are a total of 12 possible strategies (3 choices from node d1 × 2 choices at node d2 × 2 choices at node d3). But some of these are not unique. For example if the choice at node d1 is “sell”, the choices at nodes d2 and d3 (4 possible combinations) are unimportant; all four such strategies are identical.

Method evaluate calculates the expected cost, benefit and utility of each traversable path for each strategy, and aggregates by strategy. The results for the gas problem are computed as follows. Pay-off is defined as benefit minus cost.

# find optimal strategies
RES <- DT$evaluate()
RES[, "Payoff"] <- RES[, "Benefit"] - RES[, "Cost"]

This gives the following pay-off for each strategy:

#>      d1   d2   d3 Cost Benefit Payoff
#> 1   dig  dig  dig  300    1750   1450
#> 2   dig  dig sell  300    1750   1450
#> 3   dig sell  dig  300    1750   1450
#> 4   dig sell sell  300    1750   1450
#> 5  sell  dig  dig    0     800    800
#> 6  sell  dig sell    0     800    800
#> 7  sell sell  dig    0     800    800
#> 8  sell sell sell    0     800    800
#> 9  test  dig  dig  350    1750   1400
#> 10 test  dig sell  134     895    761
#> 11 test sell  dig  266    1743   1477
#> 12 test sell sell   50     888    838

The optimal strategy is test;sell;dig, i.e., test, sell if negative and dig otherwise. The expected pay-off from this strategy is 1477.