Desktop version

Home arrow Mathematics

  • Increase font
  • Decrease font


<<   CONTENTS   >>

Examples of Partial Conflict Games

We present examples of partial conflict games with their solutions.

Example 7.9. Cuban Missile Crisis—A Classic Game of Chicken[1].

“We’re eyeball to eyeball, and I think the other fellow just blinked,” were the eerie words of Secretary of State Dean Rusk at the height of the Cuban Missile Crisis in October, 1962. Secretary Rusk was referring to signals by the Soviet Union that it desired to defuse the most dangerous nuclear confrontation ever to occur between the superpowers, which many analysts have interpreted as a classic instance of a nuclear “Game of Chicken.”

We will highlight the scenario from 1962. The Cuban Missile Crisis was precipitated in October 1962, by the Soviet’s attempt to install medium- and intermediate-range nuclear-armed ballistic missiles in Cuba that were capable of hitting a large portion of the United States. The range of the missiles from Cuba allowed for major political, population, and economic centers to become targets. The goal of the United States was immediate removal of the Soviet missiles. U.S. policy makers seriously considered two strategies to achieve this end: naval blockade or airstrikes.

President Kennedy, in his speech to the nation, explained the situation as well as the goals for the United States. He set several initial steps. First, to halt the offensive build-up, a strict quarantine on all offensive military equipment under shipment to Cuba was being initiated. He went on to say that any launch of missiles from Cuba at anyone would be considered an act of war by the Soviet Union against the United States resulting in a full retaliatory nuclear strike against the Soviet Union. He called upon Soviet Premier Krushchev to end this threat to the world, and restore world peace.

We will use the Cuban Missile Crisis to illustrate parts of the theory— not just an abstract mathematical model, but one that mirrors the real-life choices, and underlying thinking of flesh-and-bloocl decision makers. Indeed, Theodore Sorensen, special counsel to President Kennedy, used the language of “moves” to describe the deliberations of EXCOM, the Executive Committee of key advisors to President Kennedy during the Cuban Missile Crisis. Sorensen said,

We discussed what the Soviet reaction would be to any possible move by the United States, what our reaction with them would have to be to that Soviet action, and so on, trying to follow each of those roads to their ultimate conclusion.

Problem: Build a mathematical model that allows for consideration of alternative decisions by the two opponents.

Assumption: We assume the two opponents are rational players.

Model Choice: Game Theory.

The goal of the United States was the immediate removal of the Soviet missiles; U.S. policy makers seriously considered two strategies to achieve this end:

  • 1. .4 naval blockade (B), or "quarantine" as it was euphemistically called, to prevent shipment of more missiles, possibly followed by stronger action to induce the Soviet Union to withdraw the missiles already installed.
  • 2. .4 'surgical" air strike (A) to destroy the missiles already installed, insofar as possible, perhaps followed by an invasion of the island.

The alternatives open to Soviet policy makers were:

  • 1. Withdrawal (W) of their missiles.
  • 2. Maintenance (M) of their missiles.

We set (ж, у) as (payoffs to the United States, payoffs to the Soviet Union) where 4 is the best result, 3 is next best, 2 is next worst, and 1 is the worst. Table 7.48 shows the payoffs.

TABLE 7.48: Cuban Missile Crisis Payoffs

Soviet Union

Withdraw

Missiles

(W)

Maintain

Missiles

(M)

United

States

Blockade (B)

(3,3)

(2,4)

Air Strike (A)

(4,2)

(1,1)

We show the movement diagram in Table 7.49 where we have equilibria at

(4.2) and (2,4). The Nash equilibria are boxed. Note both equilibria, (4,2) and (2,4) are found by our arrow diagram.

As in Chicken, as both players attempt to get to their equilibrium, the outcome of the games end up at (1,1). This is disastrous for both countries and their leaders. The best solution is the (3,3) compromise position. However,

(3.3) not stable. This choice will eventually put us back at (1,1). In this situation, one way to avoid the “chicken dilemma” is to try strategic moves.

Both sides did not choose their strategies simultaneously or independently. Soviets responded to our blockade after it was imposed. The U.S. held out the chance of an air strike as a viable choice even after the blockade. If the U.S.S.R. would agree to remove the weapons from Cuba, the U.S. would agree

TABLE 7.49: Cuban Missile Crisis Movement Diagram

Soviet Union

Withdraw

Missiles

(W)

Maintain

Missiles

(M)

Rose

Blockade (В)

Air Strike (A)

to (a) remove the quarantine and (b) agree not to invade Cuba. If the Soviets maintained their missiles, the U.S. preferred the airstrike to the blockade. Attorney General Robert Kennedy said, “if they do not remove the missiles, then we will.” The U.S. used a combination of promises and threats. The Soviets knew our credibility in both areas was high (strong resolve). Therefore, they withdrew the missiles, and the crisis ended. Khrushchev and Kennedy were wise.

Needless to say, the strategy choices, probable outcomes, and associated payoffs shown in Table 7.48 provide only a skeletal picture of the crisis as it developed over a period of thirteen days. Both sides considered more than the two alternatives listed, as well as several variations on each. The Soviets, for example, demanded withdrawal of American missiles from Turkey as a quid pro quo for withdrawal of their own missiles from Cuba, a demand publicly ignored by the United States.

Nevertheless, most observers of this crisis believe that the two superpowers were on a collision course, which is actually the title of one book[2] describing this nuclear confrontation. Analysts also agree that neither side was eager to take any irreversible step, such as one of the drivers in Chicken might do by defiantly ripping off the steering wheel in full view of the other driver, thereby foreclosing the option of swerving.

Although in one sense the United States “won” by getting the Soviets to withdraw their missiles, Premier Nikita Khrushchev of the Soviet Union at the same time extracted from President Kennedy a promise not to invade Cuba, which seems to indicate that the eventual outcome was a compromise of sorts. But this is not game theory’s prediction for Chicken because the strategies associated with compromise do not constitute a Nash equilibrium.

To see this, assume play is at the compromise position (3,3), that is, the U.S. blockades Cuba, and the U.S.S.R. withdraws its missiles. This strategy is not stable because both players would have an incentive to defect to their more belligerent strategy. If the U.S. were to defect by changing its strategy to airstrike, play would move to (4. 2), improving the payoff the U.S. received; if the U.S.S.R. were to defect by changing its strategy to Maintenance, play would move to (2,4), giving the U.S.S.R. a payoff of 4. (This classic game theory setup gives us no information about which outcome would be chosen, because the table of payoffs is symmetric for the two players. This is a frequent problem in interpreting the results of a game theoretic analysis, where more than one equilibrium position can arise.) Finally, should the players be at the mutually worst outcome of (1,1), that is, nuclear war, both would obviously desire to move away from it, making the strategies associated with (1,1), like those with (3,3), unstable.

Example 7.10. Writer’s Guild Strike of 2007-2008.

Game Theory Approach[3]

Let us begin by stating strategies for each side. Our two rational players will be the Writer ’s Guild and the Management. We develop strategies for each player.

Strategies:

  • • Writer’s Guild: Their strategies are to strike (S) or not to strike (NS).
  • • Management : Salary Increase and revenue sharing (IN) or status quo (SQ).

First, we rank order the outcomes for each side in order of preference. (The rank orderings are ordinal utilities.)

Alternatives and Rankings

• Strike vs. Status Quo = (S, SQ):

W riter’s worst case (1); Management’s next to best case (3)

• No Strike vs. Status Quo = (NS, SQ):

Writer’s next to worst case (2); Management’s best case (4)

• Strike vs. Salary Increase and Revenue Sharing = (S, IN):

Writers next to best case (3); Management’s next to worst case (2)

• No Strike vs. Salary Increase and Revenue Sharing = (NS, IN):

Writer’s best case (4); Management’s worst case (1)

This list provides us with a payoff matrix consisting of ordinal values; see Table 7.50. We will refer to the Writer’s Guild as Rose and the Management as Colin.

The movement diagram in Table 7.51 finds (2,4) as the likely outcome.

TABLE 7.50: Payoff Matrix for Writers and Management

Management

(Colin)

Status Quo

(SQ)

Increase Salary (IN)

Writer’s

Guild

(Rose)

Strike (S)

(1,3)

(3,2)

No Strike (NS)

(2,4)

(4,1)

TABLE 7.51: Writer’s Guild and Management’s Movement Diagram

Management (Colin)

Status Quo

(SQ)

Increase

Salary

(IN)

Writer’s

Guild

(Rose)

Strike (S)

No Strike (NS)

The movement arrows point towards (2,4) as the pure Nash equilibrium. We also note that this result is not satisfying to the Writer’s Guild, and that they would like to have a better outcome. Both (3,2) and (4,1) within the payoff matrix provide better outcomes to the Writers.

The Writers can employ several options to try to secure a better outcome. They can first try Strategic Moves, and if that fails to produce a better outcome, then they can move to Nash Arbitration. Both of these methods employ communications in the game. In strategic moves, we examine the game to see if the outcome is changed by “moving first”, threatening our opponent, or making promises to our opponent, or whether a combination of threats and promises changes the outcome.

Examine the strategic moves. If the writers move first; their best result is again (2,4). If management moves first, the best result is (2,4). First moves keep us at the Nash equilibrium. The writers consider a threat: they tell management that if they choose SQ, they will strike putting us at (1,3). This result is indeed a threat, as it is worse for both the writers and management. However, the options for management under IN are both worse than (1,3), so they do not accept the threat. The writers do not have a promise to offer. At this point we might involve an arbiter using the Nash arbitration method as suggested earlier.

The Nash arbitration formulation then is

Writers and management security levels are found from prudential strategies using Equations (7.9) and (7.10). The security levels are calculated to be

(2,3). We show this in Figure 7.4.

The Payoff Polygon for Writer’s Guild Strike

FIGURE 7.4: The Payoff Polygon for Writer’s Guild Strike

The Nash equilibrium value (2,4) lies along the Pareto Optimal line segment (from (4,1) to (2,4)). But the Writers can do better by going on strike and forcing arbitration, which is what they did.

In this example, we consider “binding arbitration” where the players have a third party work out the outcomes that best meet their desires and is acceptable to all players. Nash found that this outcome can be obtained by:

The status quo point is formed from the security levels of each side.

We find the value (2, 3) using prudential strategies. The function for the Nash Arbitration scheme is Maximize (x — 2){y — 3). Using Maple’s QPSolve from the Optimization package, we find the desired solution to our quadratic program is x = 2.33 and у = 3.5.

We have the (x,y) = (2.33,3.5) as our arbitrated solution. We can now determine how the arbiters should proceed. We solve the following simultaneous equations

We find that the probabilities to be played are 5/6 and 1/6. Further, we see that Player I, the Writers, should always play II2, so the management arbiter plays 5/6 • C and 1/6 • C2 during the arbitration.

Example 7.11. Game Theory Applied to a Dark Money Network.

“Dark money” originally referred to funds given to nonprofit organizations who can spend the money to influence elections and policy while not having to disclose their donors.[4] The term has extended to encompass nefarious groups seeking to undermine a government. Discussing the strategies for defeating a Dark Money Network (DMN) leads naturally into the Game Theory analysis of the strategies for the DMN and for the State trying to defeat the DMN. When conducting this game theory analysis, we originally limited the analysis by using ordinal scaling, and ranking each of the four strategic options one through four. The game was set up in Table 7.52. Strategy A is for the state to pursue a non-kinetic strategy, В is a kinetic strategy. Strategy C is for the DMN to maintain its organization and D is for it to decentralize.

TABLE 7.52: Dark Money Network

Payoff

Tableau

DMN

C

D

State

A

(2,1)

(4,2)

В

(1,4)

(3,3)

This ordinal scaling worked when allowing communications and strategic moves; however, without a way of determining interval scaling it was impossible to conduct analysis of prudential strategies, Nash Arbitration, or Nash’s equalizing strategies. Here we show an application of Analytic Hierarchy Process (AHP) (See Chapter 8) in order to determine the interval scaled payoffs of each strategy for both the DMN and the State. We will use Saaty’s standard nine point preference in the pairwise comparison of combined strategies. For the State’s evaluation criteria, we chose four possible outcomes: how well the strategy degraded the DMN, how well it maintained the state’s own ability to raise funds, how well the strategy would rally their base, and finally how well it removed nodes from the DMN. The evaluation criteria we chose for the DMN’s four possible outcomes were: how anonymity was maintained, how much money the outcome would raise, and finally how well the DMN’s leaders could maintain control of the network.

After conducting an AHP analysis, we obtained a new payoff matrix in Table 7.53 with cardinal utility values.

TABLE 7.53: Dark Money Network Revised

Payoff

Tableau

DMN

C

D

State

A

(0.168,0.089)

(0.366,0.099)

В

(0.140,0.462)

(0.323,0.348)

With the cardinal scaling, it is now possible to conduct a proper analysis that might include mixed strategies or arbitrated results such as finding prudential strategies, Nash’s Equalizing Strategies, and Nash Arbitration. Using a series of game theory solvers developed by Feix[5] we obtain the following results.

  • • Nash Equilibrium: A pure strategy Nash equilibrium was found at (A,D) of (0.366,0.099) using strategies of Non-Kinetic and Decentralize.
  • • Mixed Nash Equalizing Strategies: The State plays Non-Kinetic 91.9% of the time and Kinetic 8.1% of the time; DMN always plays Maintain Organization.
  • • Prudential Strategies: The Security Levels are (0.168,0.099) when the State plays C and the DMN plays A.

Since there is no equalizing strategy for the DMN, should the State attempt to equalize the DMN. The result is as follows. This is a significant departure from our original analysis prior to including the AHP pairwise comparisons. The recommendations for the state were to use a kinetic strategy 50% of the time and a non-kinetic strategy 50% of the time. However, it is obvious that with proper scaling the recommendation should have been to execute a non- kinetic strategy the vast majority, 92%, of the time, and only occasionally, 8%, conduct kinetic targeting of network nodes. This greatly reinforces the recommendation to execute a non-kinetic strategy to defeat the DMN.

Finally, if the State and the DMN could enter into arbitration, the result would be at BD, which was the same prediction as before the proper scaling. 11

Example 7.12. Course of Actions Decision Process for Partial Conflict Games.

Return to the zero-sum game of Example 7.8 of Section 7.3. In many real world analyses, a player winning is not necessarily another player losing. Both might win or both might lose based upon the mission and courses of action played. Again, assume Player I has four strategies and Player II has six strategies that they can play. We use an AHP approach (see Fox [Fox2014]) to obtain cardinal values of the payoff to each player. Although this example is based on combat courses of action, this methodology can be used when competing players both have courses of action that they could employ. As in Example 7.11, we use AHP to convert the ordinal rankings of the combined COAs in order to obtain cardinal values.

The game’s payoff matrices become

The solutions, depending on the starting conditions for P and Q, are found using Maple’s QPSolve since Objective is quadratic or NLPSolve since it is nonlinear. As before, use fnormal to eliminate numerical artifacts.

We find Player I should play a pure strategy' of COA 4, while Player 2 should play either a mixed strategy of y5 = 8/11 and yG = 3/11, or a pure strategy of always у4. We also employed sensitivity analysis by varying the criteria weights which represent the cardinal values. We found not much change in the results from our solution analysis presented here.

Exercises

Solve the problems using any method.

1. Find all the solutions for Table 7.54.

TABLE 7.54: U.S. vs. State-Sponsored Terrorism

Payoff

Tableau

State Sponsor

Sponsor

Terrorism

Stop Sponsoring Terrorism

U.S.

Strike Militarily

(2M)

(1.5,0)

Do Not Strike Militarily

(1,1.5)

(0,4)

2. Find all the solutions for Table 7.55.

TABLE 7.55: U.S. vs. State-Sponsored Terrorism

Payoff

Tableau

State Sponsor

Sponsor

Terrorism

Stop Sponsoring Terrorism

U.S.

Strike Militarily

(3,5)

(2.1)

3. Find all the solutions for Table 7.56.

TABLE 7.56: U.S. vs. State-Sponsored Terrorism

Payoff

Tableau

Colin

Arm

Disarm

Rose

Arm

(2,2)

(1,1)

Disarm

(4Л)

(3,3)

  • 4. Consider the following classical games of Chicken in Tables 7.57 to 7.59. Find the solutions.
  • (a)

TABLE 7.57: A Classical Game of Chicken I

Payoff

Tableau

Colin

C

D

Rose

A

(3,3)

(2,4)

В

(4,2)

(1,1)

(b)

TABLE 7.58: A Classical Game of Chicken II

Payoff

Tableau

Colin

C

D

Rose

A

(2,3)

(4,1)

В

(1,2)

(3,4)

(c)

TABLE 7.59: A Classical Game of Chicken III

Payoff

Tableau

Colin

C

D

Rose

A

(4,3)

(3,4)

В

(2,1)

(1,2)

  • 5. Consider the classical games of Prisoner’s Dilemma in Tables 7.60 and 7.61. Find the solutions.
  • (a)

TABLE 7.60: A Classical Game of Prisoner’s Dilemma I

Payoff

Tableau

Colin

C

D

Rose

A

(3,3)

(-1,5)

В

(5,-1)

(0,0)

(b)

TABLE 7.61: A Classical Game of Prisoner’s Dilemma II

Payoff

Tableau

Colin

C

D

Rose

A

(3,3)

(1,5)

В

(4,0)

(0,2)

Projects

Project 7.1. Corporation XYZ consists of Companies Rose and Colin. Company Rose can make Products R i and R2. Company Colin can make Products Ci and С-2- These products are not in strict competition with one another, but there is an interactive effect depending on which products are on the market at the same time as reflected in Table 7.62 below. The table reflects profits in millions of dollars per year. For example, if products R2 and C are produced and marketed simultaneously, Rose’s profits are 4 million and Colin’s 5 million annually. Rose can make any mix of R and R2, and Colin can make any mix of C and C'2. Assume the information below is known to each company.

NOTE: The CEO is not satisfied with just summing the total profits. He might want the Nash Arbitration Point to award each company proportionately based on their strategic positions, if other options fail to produce the results he desires. Further, he does not believe a dollar to Rose has the same importance to the corporation as a dollar to Colin.

TABLE 7.62: Corporate Payoff Matrix

Payoff

Tableau

Company Colin

Ci

c2

Company

Rose

Ri

(3,7)

(8,5)

R-2

(4,5)

(5,6)

a. Suppose the companies have perfect knowledge and implement market strategies independently without communicating with one another. What are the likely outcomes? Justify your choice.

b. Suppose each company has the opportunity to exercise a strategic move. Try first moves for each player; determine if a first move improves the results of the game.

c. In the event things turn “hostile” between Rose and Colin, find, state, and then interpret

i. Rose’s Security Level and Prudential Strategy'.

ii. Colin’s Security Level and Prudential Strategy.

Now suppose that the CEO is disappointed with the lack of spontaneous cooperation between Rose and Colin, and decides to intervene and dictate the “best” solution for the corporation. The CEO employs an arbiter to determine an “optimal production and marketing schedule” for the corporation. What is this strategy? d. Explain the concept of “Pareto Optimal” from the CEO’s point of view. Is the “likely outcome” you found in question (1) at or above Pareto Optimal? Briefly explain and provide a payoff polygon plot.

e. Find and state the Nash Arbitration Point using the security levels found above.

f. Briefly discuss how you would implement the Nash Point. In particular, what mix of the products R and i?2 should Rose produce and market, and what mix of the products C and C'2 should Colin produce? Must their efforts be coordinated, or do they simply need to produce the “optimal mix”? Explain briefly.

g. How much annual profit will Rose and Colin each make when the CEO’s dictated solution is implemented?

  • [1] Adaptecl from Brams, “Game theory and the Cuban missile crisis,” available athttp://plus.maths.org/issuel3/features/brams/index.html.
  • [2] Henry M. Pachter, Collision Course: The Cuban Missile Crisis and Coexistence,Praeger, NY, 1963.
  • [3] From [Fox2008].
  • [4] ’“Adapted from Couch et al. [CFE2016].
  • [5] Feix, Game Theory: Toolkit and Workbook for Defense Analysis Students, DefenseTech. Info. Center, 2007. Available at http://www.dtic.mil/dtic/tr/fulltext/u2/a470073.pdf.
 
<<   CONTENTS   >>

Related topics