1. How Much and On What? Defending and Deterring Strategic Attackers
Robert Powell
Travers Department of Political Science
UC Berkeley

2. The Strategic Allocation of Limited Resources
An old question in game theory.
Colonel Blotto Games
Zero-sum, simultaneous-move games in which two players allocate resources across a known number of sites.
Borel 1921, Tukey 1949, Blackett 1958, Shubik and Weber 1981, Golman and Page 2006, Roberson 2006.
All-pay auctions
Kvasov 2006
The basic issue is what might be called the inefficiency puzzle of war or more generally the use of power.
Recent formal work sees the causes and conduct of war in terms of what might be called an inefficiency puzzle.
To frame the problem, suppose that a group of actors are bargaining about how to resolve an issue or, more abstractly, about how to divide a "pie." One or more of them can affect the outcome and possibly even impose a division through the use of some form of power -- be it military, economic, legal, or more generally political. The exercise of power, however, consumes resources, and, consequently, the pie to be divided among the bargainers before anyone tries to impose a settlement is larger than it will be afterward. As a result, there usually are divisions of the larger pie that would have given each bargainer more than it will obtain if anyone does try to impose a settlement. The use of power, in other words, leads to Pareto inefficient outcomes. Why, then, do the bargainers fail to reach a Pareto superior agreement prior to the explicit use of power?
Why bargaining breaks down in inefficient outcomes is of course an old question in economics. Framing the problem of war this way makes it possible to look to other theoretical and empirical work on the efficiency in the hope of getting some leverage on the problem of war.

3. The Strategic Allocation of Limited Resources
An old question in game theory.
Colonel Blotto Games
All-pay auctions
Renewed Attention After the Attacks of 9/11
Secretary Chertoff’s summary:
“Although we have substantial resources to provide security, these resources are not unlimited. Therefore, as a nation, we must make tough choices about how to invest finite human and financial capital to attain the optimal state of preparedness.”
The basic issue is what might be called the inefficiency puzzle of war or more generally the use of power.
Recent formal work sees the causes and conduct of war in terms of what might be called an inefficiency puzzle.
To frame the problem, suppose that a group of actors are bargaining about how to resolve an issue or, more abstractly, about how to divide a "pie." One or more of them can affect the outcome and possibly even impose a division through the use of some form of power -- be it military, economic, legal, or more generally political. The exercise of power, however, consumes resources, and, consequently, the pie to be divided among the bargainers before anyone tries to impose a settlement is larger than it will be afterward. As a result, there usually are divisions of the larger pie that would have given each bargainer more than it will obtain if anyone does try to impose a settlement. The use of power, in other words, leads to Pareto inefficient outcomes. Why, then, do the bargainers fail to reach a Pareto superior agreement prior to the explicit use of power?
Why bargaining breaks down in inefficient outcomes is of course an old question in economics. Framing the problem of war this way makes it possible to look to other theoretical and empirical work on the efficiency in the hope of getting some leverage on the problem of war.

4. Game Theory vis-à-vis Decision Theory
When considering an adversary’s possible courses of action…
Decision theory postulates the probabilities (based on experience and evidence outside the decision model).
Game theory tries to infer these probabilities
Intent (probability of using attack mode x to strike target y) should come out of the analysis, not be assumed as an input to the analysis.
The basic issue is what might be called the inefficiency puzzle of war or more generally the use of power.
Recent formal work sees the causes and conduct of war in terms of what might be called an inefficiency puzzle.
To frame the problem, suppose that a group of actors are bargaining about how to resolve an issue or, more abstractly, about how to divide a "pie." One or more of them can affect the outcome and possibly even impose a division through the use of some form of power -- be it military, economic, legal, or more generally political. The exercise of power, however, consumes resources, and, consequently, the pie to be divided among the bargainers before anyone tries to impose a settlement is larger than it will be afterward. As a result, there usually are divisions of the larger pie that would have given each bargainer more than it will obtain if anyone does try to impose a settlement. The use of power, in other words, leads to Pareto inefficient outcomes. Why, then, do the bargainers fail to reach a Pareto superior agreement prior to the explicit use of power?
Why bargaining breaks down in inefficient outcomes is of course an old question in economics. Framing the problem of war this way makes it possible to look to other theoretical and empirical work on the efficiency in the hope of getting some leverage on the problem of war.

5. Goal of the Paper
Existing work generally takes the defender’s budget as given as in the classic Blotto game.
Today’s paper lets the defender determine how much to it wants to spend as well as how to allocate the resources it does spend.
Perhaps more interestingly, we can also think about the effects the defender’s spending has on the attacker’s effort. In principle, the defender can fully deter the attacker, i.e., induce it to exert zero effort in attacking the defender.
The paper I want to present today builds on some of my prior work. In all of that work, the defender’s budget was an exogenous parameter as in the classic blotto game.

6. Four Key Results
The optimal (equilibrium) allocation of the defender's resources minmaxes the attacker.
Decompose the gain from increased spending into the sum of a defensive effect, a deterrent effect, and a cost effect.
The defender's allocation and level-of-spending problems are separable.
Facing a more determined attacker has no effect on the way the defender and attacker allocate their spending and effort, but does lead to greater levels of spending
The paper I want to present today builds on some of my prior work. In all of that work, the defender’s budget was an exogenous parameter as in the classic blotto game.

7. Plan of the Talk
Describe the model with endogenous levels of defender spending and attacker effort.
Give key intuition in a setting with a fixed budget.
Sketch the analysis when the defender and attacker determine the levels of spending and effort.
The paper I want to present today builds on some of my prior work. In all of that work, the defender’s budget was an exogenous parameter as in the classic blotto game.

8. The General Model N sites
The defender decides how much to spend on protecting site j, rj , with total spending given by R =  rj .
After observing the defender’s allocation, the attacker decides how much effort ej to invest in attacking site j with total effort given by E =  ej .
The defender suffers a loss j > 0 and the attacker gains j > 0 if site j is destroyed. Both get zero if the site is not destroyed.
Costs of allocating R and E , cD (R ) and cA (E ), are increasing and convex.
The probability that j is destroyed if attacked, Vj (rj,ej ), is decreasing and weakly convex in spending and increasing in effort.

9. Simplifying Assumption
Substantive:
The marginal effect of additional effort on the vulnerability of a site is independent of the level of effort already being exerted.
To Make the Algebra Easier:
All sites are identical:  = j,  = j, v (r ) = vj(r )
v (r ) = 1 – vr
cD (R) = kD R 2 and cA (E) = kA E 2 / 2.
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10. The Payoffs
The defender’s losses and attacker’s gain are given by:

11. The Effects of Spending More

12. A Simple Allocation Rule – Even with Different Kinds of Sites

13. Optimal Allocation with Identical Sites
The optimal allocation equates the attacker’s expected payoffs to striking the various sites.
If, to simplify matters, all of the sites are identical, then each site gets the same amount
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14. Solving the Game: The Attacker’s Problem
Given that the defender spends R, the attacker’s payoff is:
The optimal level of effort is:
Intuitive check:
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15. Solving the Game: The Defender’s Problem
Given that the attacker exerts E* (R) in response to R, the defender’s losses are:
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16. Determining the Optimal Level of Spending
The optimal level of spending occurs where the marginal reduction in losses from spending more is zero: ∂L/∂R = 0.
The marginal effect of increased spending can be decomposed into the sum of a defensive effect, a deterrent effect, and a cost effect.
I was aware of the limitation a

17. The Subgame Perfect Equilibrium Allocations
Tedious algebra gives:
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18. Conclusion: Four Key Results
The optimal (equilibrium) allocation of the defender's resources minmaxes the attacker.
The marginal effect of increased spending can be decomposed into the sum of a defensive effect, a deterrent effect, and a cost effect.
The defender's allocation and level-of-spending problems are separable.
Facing a more determined attacker has no effect on the way the defender and attacker allocate their spending and effort, but does lead to greater levels of spending
The paper I want to present today builds on some of my prior work. In all of that work, the defender’s budget was an exogenous parameter as in the classic blotto game.