### 9.2.1 Single-Stage Decision Networks

The decision tree is a state-based representation because the worlds correspond to the resulting state. It is, however, often more natural and more efficient to represent and reason in terms of features, represented as variables.

A **single-stage decision network** is an extension of a belief network
that has three kinds of nodes:

**Decision nodes**, drawn as rectangles, represent decision variables. The agent gets to choose a value for each decision variable. Where there are multiple decision variables, we assume there is a total ordering of the decision nodes, and the decision nodes before a decision node*D*in the total ordering are the parents of*D*.**Chance nodes**, drawn as ovals, represent random variables. These are the same as the nodes in a belief network. Each chance node has an associated domain and a conditional probability of the variable, given its parents. As in a belief network, the parents of a chance node represent conditional dependence: a variable is independent of its non-descendants, given its parents. In a decision network, both chance nodes and decision nodes can be parents of a chance node.- A
**utility node**, drawn as a diamond, represents the utility. The parents of the utility node are the variables on which the utility depends. Both chance nodes and decision nodes can be parents of the utility node.

Each chance variable and each decision variable has an associated domain. There is no domain associated with the utility node. Whereas the chance nodes represent random variables and the decision nodes represent decision variables, there are no associated utility variables. The utility provides a function of its parents.

Associated with a decision network is a conditional probability for each chance node given its parents (as in a belief network) and a representation of the utility as a function of the utility node's parents. In the specification of the network, there are no tables associated with the decision nodes.

Which Way | Accident | Value |

short | true | 0.2 |

short | false | 0.8 |

long | true | 0.01 |

long | false | 0.99 |

Wear Pads | Which Way | Accident | Utility |

true | short | true | 35 |

true | short | false | 95 |

true | long | true | 30 |

true | long | false | 75 |

false | short | true | 3 |

false | short | false | 100 |

false | long | true | 0 |

false | long | false | 80 |

**Example 9.8:**Figure 9.5 gives a decision network representation of Example 9.4. There are two decisions to be made: which way to go and whether to wear padding. Whether the agent has an accident only depends on which way they go. The utility depends on all three variables.

This network requires two factors: a factor
representing the conditional
probability, *P(Accident|WhichWay)*,
and a factor representing the utility as a function of
*Which Way*,
*Accident*, and *Wear Pads*. Tables for these factors are shown in Figure 9.5.

A **policy** for a single-stage decision network is an assignment
of a value to each decision variable. Each policy has an **expected
utility**, which is the conditional expected
value of the utility conditioned on the
policy.
An **optimal policy** is a
policy whose expected utility is maximal. That is, it is a policy such
that no other policy has a higher expected utility.

**Procedure**OptimizeSSDN(

*DN*)

2:

**Inputs**

3:

*DN*a single stage decision network

4:

**Output**

5: An optimal policy and the expected utility of that policy.

6: Prune all nodes that are not ancestors of the utility node.

7: Sum out all chance nodes.

8: - at this stage there is a single factor

*F*that was derived from utility

9: Let

*v*be the maximum value in

*F*

10: Let

*d*be an assignment that gives the maximum value

11: return

*d,v*

Figure 9.6 shows how **variable elimination**
can be used to find an optimal policy in a
single-stage decision network. After pruning irrelevant
nodes and summing out all random variables, there will be a single
factor that represents the expected utility for each combination of
decision variables. This factor does not have to be a factor on
*all* of the decision variables; however, those decision variables that
are not included are not relevant to the decision.

**Example 9.9:**Consider running

*OptimizeSSDN*on the decision network of Figure 9.5. No nodes can be pruned, so it sums out the only random variable,

*Accident*. To do this, it multiplies both factors because they both contain

*Accident*, and sums out

*Accident*, giving the following factor:

Wear Pads | Which Way | Value |

true | short | 0.2* 35+ 0.8*95=83 |

true | long | 0.01*30+0.99*75=74.55 |

false | short | 0.2*3+0.8*100=80.6 |

false | long | 0.01*0+0.99*80=79.2 |

Thus, the policy with the maximum value - the optimal policy - is to take the short way and wear pads, with an expected utility of 83.