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#### 2.GAME THEORY FOR SMALLPOX

The smallpox debate in the United States has focused upon three kinds of attack and four kinds of defense. The three attack scenarios suppose that there might be:

• no smallpox attack

• a lone terrorist attack on a small area (similar to the likely scenario for the anthrax letters)

• a coordinated terrorist attack upon multiple population centers.

The four defense scenarios that have been publicly considered by United States agency officials are:

• stockpile smallpox vaccine

• stockpile vaccine and develop biosurveillance capabilities

• stockpile vaccine, develop biosurveillance, and inoculate key personnel

• provide mass vaccination to non-immunocompromised citizens in advance.

Although there are many refinements that can be considered for both the attack and the defense scenarios, these represent the possibilities discussed in the public meetings held in May and June 2002 (McKenna, 2002).

Suppose that analysts used game theory as one tool to evaluate potential defense strategies. Then the three kinds of attack and four kinds of defense determine a classic normal-form payoff matrix for the game [see Table 1].

The Cij entries are the costs (or payoffs) associated with each combination of attack and defense, and we have used abbreviated row and column labels to identify the defenses and attacks, respectively, as described before.

For each of the 12 attack-defense combinations, there is an associated cost. These costs may include dollars, human lives, time, and other resources. For our calculation, all of these costs are monetized, according to principles detailed in Section 3. And the monetized value of a human life is set to \$750,000, following the Department of Transportation’s human capital model that estimates value from average lost productivity (non-market approaches tend to give larger values).

Note that there is very large uncertainty in the Cij values. Portions of the cost (e.g., those associated with expenses

TABLE 1 Attack-Defense Cost Matrix

 No Attack Single Attack Multiple Attack Stockpile Vaccine C11 C12 C13 Biosurveillance C21 C22 C23 Key Personnel C31 C32 C33 Everyone C41 C42 C43

already entailed) may be known, but the total cost in each cell is a random variable. These random variables are not independent, since components of the total cost are common to multiple cells. Thus it is appropriate to regard the entire game theory table as a multivariate random variable whose joint distribution is required for a satisfactory analysis that propagates uncertainty in the costs through to uncertainty about best play.

Classical game theory (cf. Myerson 1991, Chapter 3) determines the optimal strategies for the antagonists via the minimax theorem. This theorem asserts that for any two-person cost matrix in a strictly competitive game (which is the situation for our example), there is an equilibrium strategy such that neither player can improve their expected payoff by adopting a different attack or defense. This equilibrium strategy may be a pure strategy, in which case optimal play is a specific attack-defense pair. This happens when the attack that maximizes the minimum damage and the defense that minimizes the maximum damage coincide in the same cell. Otherwise, the solution is a mixed strategy, in which case the antagonists pick attacks and defenses according to a probability distribution that must be calculated from the cost matrix. There may be multiple equilibria that achieve the same expected payoff, and for large matrices it can be difficult to solve the game.

Alternatively, one can use Bayesian decision theory to solve the game. Here a player puts a probability distribution over the actions of the opponent, and then chooses their own action so as to minimize the expected cost (cf. Myerson 1991, Chapter 2). Essentially, one just multiplies the cost in each row by the corresponding probability, sums these by row, and picks the defense with the smallest sum. This formulation is easier to solve, but it requires one to know or approximate the opponent’s probability distribution and it does not take full account of the mutual strategic aspects of adversarial games (i.e., the assigned probabilities need not correspond to any kind of “if I do this then he’ll do that” reasoning). Bayesian methods are often used in extensive-form games, where players make their choices over time, conditional on the actions of their opponent.

In developing our analysis of the smallpox example we make two assumptions about time. First, we use only the information available by June 1, 2002; subsequent information on the emerging program costs is not included. This keeps the analysis faithful in spirit to the decision problem actually faced by U.S. government policy makers in the spring of 2002 (their initial plan was universal vaccination, but ultimately they chose the third scenario with stockpiling, biosurveillance, and very limited vaccination of some first responders). Second, all of the estimated cost forecasts run to October 1, 2007. The likelihood of changing geopolitical circumstances makes it unrealistic to attempt cost estimates beyond that fiscal year.

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