Game Theory

 

 

 

 

V. Decision Graphs and Sequential Decisions

 

 

C1-C5 are called terminal nodes, those are where payoffs are assigned. Branches are called edges and where edges separate are called nodes.

 

Optimal Decision path- path that leads to optimal payoff.

 

Method of Backwards Induction (Example using above graph)

 

How can I choose to go to college or not, well look at the terminal nodes and pick the optimal ones. So if I do go to college I can then choose C1 (70,000), or no college C4 (25,000).  So my choice is simplified by assuming that ill make the best choice in the future. Now clearly my best choice in the present is to go to college.

 

Minimizing Cost using Backwards Induction.

 

Then we must find the minimal cost (sum) from getting from the beginning to a terminal node (for each step go back one node and cutoff each branch that’s sum is greater and you will get the minimal cost at the far right path with sum 5.)

 

 

Problem Set 6 :

 

A manufacturer has a product that goes through four steps A-D. Find how they could minimize there cost of production per unit.

 

 

 

 

 

 

 

End of Problem Set 6

 

Uncertainty and Single-Person Decisions [Ali, 89]

 

Pharmaceutical Company (X) , making successive decisions in whether to invest high (HI) or low (LOW) amounts of capital into developing a drug and then given a successful development whether to market (M or DM) that drug.

 

Now the Company must make a choice, whether or not to make a hi or low investment into the development of this drug. This decision graph will help them make such a choice.

For they see that

 

Exp(HI)=p(750,000) + (1-p)(-150,000)

Exp(Lo)=q(1,000,000) + (1-q)(-10,000)

 

When Exp(Hi)=Exp(Lo) these are called indifference points, which occur about when

P=1.12q + .16 this is when we don’t really care whether or not we invest hi or lo but when

 

 

Problem Set 7 :

Sequential Three Person Games

 

Use Backwards Inductions to solve this decision graph with 3 players. Where (x,y,z) indicates that x is the payoff for P1 and y,z for P2,P3 likewise. Also the decision is at the nodes is left to the player noted, i.e. P2.

 

 

 

End of Problem Set 7

 

 

 

 

 

 

References and Further Reading:

1.C. D. Aliprantis, and S. K. Chakrabarti, Games and Decision Making, Oxford   

   University Press, New York, 2000.

2. P. Straffin, Game Theory and Strategy, The Mathematical Association of America, DC,

   1993.