Game theory

The concepts and tools of game theory is a branch of microeconomics. Game theory has been widely used not only in business but also to analyze the effects of selecting alternative strategies to achieve a military objective.

2.4.1 The concept of game theory

For games of opposed interests, the basic concepts of maxmin and equilibrium strategies are defined and illustrated. Moving to general noncooperative games, the concepts of Stackelberg equilibrium and disequilibrium are presented in a duopoly game, and two logically consistent foundations for the competitive solution are given. The credibility of threats is discussed, and perfect equilibrium defined. Gaming is used by researchers interested in how people learn and play games and by other analysts interested in exploring strategies and policies, as a vehicle for helping understand complex issues [33].

People learn from gaming by designing games, playing them, or analyzing game results. Unlike many other techniques of analysis, gaming is not a solution method. The output of a good game is increased understanding. Gaming can be used along with other methods in conducting a study. Regardless of whether gaming achieves the rigor early proponents sought, it appears to have continuing value [34].

Game theory has been widely used to analyze the effects of selecting alternative strategies to achieve a military objective. In two-person zero-sum games, i.e., a payoff to player 1 is a loss to player 2, both players have several alternative strategies they may pursue and, although each is aware of the strategies available to his opponent, neither is aware of the strategy his opponent will select. Therefore each player may select a strategy that will maximize his minimum payoff. Such a player will hedge against the likelihood that his opponent will select the strategy that results in the worst payoff. The effects of knowing about an opponent’s strategy makes game theory an excellent place to start a discussion of the effects of information on combat outcomes (payoffs). We do this by allowing each of the players (actually, “sides” in a battle) to possess varying amounts of relevant information about the strategy his opponent will select, and then we measure the

effect this has on the outcome of the game [3]. In essence, we are postulating varying levels of K_B and K_R. We have designed four games in which the amount of information possessed by each side (K_B and K_R) is allowed to vary. Side 1’s information might be thought of, by analogy, as comparable to that available to the U.S. Army in

• The current force, the Army of Excellence (AOE) (Game 1);

• Army XXI (Game 2); and

• Army After Next3 (AAN) (Games 3 and 4).

In addition to four different assumptions about the information available to both sides, we considered three cases of dimensionality with respect to the number of strategies or choices available to both sides. We allow each side three, five, or ten choices. (This feature of the game has some intuitive relationship with warfare, where the value of intelligence relates to the degrees of freedom available to opposing sides, which are usually rather limited.) All the games have the structure depicted in Figure 2 have choices i = 1, 2, . . . , m and j = 1, minimize the payoff. This leads Side 1 to pursue what is referred to as a “maximin”

strategy and Side 2 to pursue a “minimax” strategy.

2.4.2 Selecting the Optimal Strategy

Side 1’s optimal strategy, i*, is found by first computing, for each of his possible choices i, the worst outcome (the outcome that would come about if Side 2 made the best choice consistent with Side 1’s having chosen i). We call that worst outcome ai, min, which is given by

Side 1’s most conservative choice,i , is the one that maximizes ^* a_i,min. That is, he chooses

His payoff will then be at least as good as a_max,min. For Side 2, we reverse the process. Side 2’s optimal strategy, j , is found by first computing, for each of his possible choices j, the ^* best outcome (the outcome that would come about if Side 1 made the best choice consistent with Side 2’s having chosen j). We call that the worst outcome, a_{max j.}. It is chooses the column for whicha_{max j.} is smallest:

( )

2.4.3 The variable knowledge cases

We might think about war abstractly as follows. In any given battle, Side 1’s choice of strategies will have some effect on the outcome, as will Side 2’s. Depending on the circumstances of battle (force ratios, terrain, etc.), the strategies may make more or less difference. How, then, do we think about the value of information? As an abstraction, we can consider a vast array of battles in which strategies have very different consequences for the outcomes. We can then ask how much value information would have, on average, over that vast array of battles. This is indeed what we have calculated. For each of 1,000 different battles we generated a payoff matrix as in Figure 2, using random numbers between 0 and 100. We then made various assumptions about how much knowledge each side had about the payoff matrix. Each side then selected strategies based on that knowledge. We did this first assuming that the sides had three strategies each; we repeated the work with five and ten strategies. In the discussions below, we refer to the payoff matrix depicted in Figure 2 as A.

• Game 1: current force (AOE) (both sides have correct information).

Side 1 and Side 2 have common and correct knowledge of all the values of the payoff

matrix A. Both sides have the same information about payoffs but are ignorant about each other’s choices. Neither has superior knowledge. This can be thought of as the case in whichK_B =K_R and Γ=1.

• Game 2: Army XXI (Side 1 has correct information and Side 2has incorrect information).

Side 1 has correct knowledge of all the values of A = A1, and Side 2 has a completely incorrect understanding of the payoff matrix. We simulate this by providing Side2 with a payoff matrix, A = A2, composed of a second set of random numbers between 0 and 100.

Therefore Side 2 will make decisions based on erroneous information. Although purely an abstraction, this could describe a situation in which Army XXI with superb information fights an enemy who not only lacks valid information but is thoroughly confused. This can be thought of as the case in which Blue (Side 1) has information superiority, i.e.,

. f1 f K and Γ K_{B R}

• Game 3: AAN (Side 1 has correct information, Side 2 has correct information, and Side 1 knows Side 2’s choice).

Side 1 and Side 2 have correct knowledge of the values of A, as in Game 1. Side 2

chooses his minimax strategy j* from the correct matrix A. Side 1, however, knows the choice Side 2 makes, and rather than choose his maximin strategy (i*), he focuses only on the payoffs corresponding to the minimax choice of Side 2 and maximizes his

payoff. This simulates the case in which Side 1 has perfect intelligence and, as a result, another kind or higher level of information superiority. Although Side 2’s basic information in this case (as opposed to Game 2) is not bad, it is clearly inferior to Side 1’s.In this case, we have again thatK_B fK_R and Γf1. , but now Γ is significantly greater than 1.

• Game 4: AAN (Side 1 has correct information, Side 2 has incorrect information, and Side 1 knows Side 2’s choice).

In the fourth game Side 1 has correct knowledge of all the values of A = A1 and Side 2 has a completely incorrect payoff matrix A = A2 composed of a second set of random numbers between 0 and 100, as in Game 2. Side 2 chooses his minimax strategy, j*, from the incorrect information in A2. Side 1 knows the choice of Side2.

Rather than using his maximin strategy, he focuses only on the payoffs corresponding to

the minimax choice of Side 2 from the incorrect information and makes his choice from the correct matrix, A1. Side 1 has perfect information (maximum knowledge). He may even have established this position by actively ensuring (through offensive information operations) that Side 2 has bad information. Thus, Side 1 enjoys not only information superiority but also information dominance, i.e., K_B fδ_B and K_B f K_R.

2.4.4 Results

Table 2 summarizes the results of the four games. In each case, three different sets of strategies, or game sizes, were involved. The entries in the table can be thought of as percentages reflecting the likelihood that Side 1 will be successful given the relative knowledge between the two sides.

Table 2. The effect of knowledge on Game Outcomes Game Size Game 1 Game 2 Game 3 Game 4

3*3 50 63 58 75

5*5 50 61 65 83

10*10 49 59 75 91

It is important to note that the table entries do not reflect the likelihood that Side 1 will experience a successful combat outcome, but rather the degree to which relative knowledge contributes to Side 1’s successful outcome: relative force ratios, weapon system effectiveness, and other measures discussed later contribute as well. A score of 90, for example, means that relative knowledge contributed 90 percent to Side 1’s successful outcome, whereas it contributed only 10 percent to Side 2’s successful outcome. The actual outcome is not of interest here, just the contribution of knowledge. The games reflect the effect of knowledge on the likelihood of a successful outcome. Beginning with Game 1, we see that, as predicted, when neither side enjoys information superiority, the likelihood of winning is even—that is, the contribution of knowledge to winning is even. This seems to hold regardless of the number of strategies available to each side. This also applies to Game 2, with Side 2 possessing erroneous information about the outcomes. The pattern appears to change, however, for Games 3 and 4. There appears to be a greater advantage to Side 1 when the number of strategies increases. This phenomenon is easy to explain based on the structure of the game. Side 2’s selections in both games approach random choices, where the probability of selecting any of the s strategies is 1/ s. Therefore, the likelihood of

succeeding is greater for smaller strategy sets. What is not clear from all this is whether the seeming advantages associated with information superiority and large strategy sets is applicable to real-world engagements. What is missing is some understanding of the relative importance of the choices being made.

2.5 The Ardennes: Battle of the Bulge

Battle of the Bulge was the story of how the high command, American and British, reacted to defeat the German plan once the reality of a German offensive was accepted.

But most of all it is the story of the American fighting man and the manner in which he fought a myriad of small defensive battles until the torrent of the German attack was slowed and diverted, its force dissipated and finally spent. It is the story of squads, platoons, companies, and even conglomerate scratch groups that fought with courage, with fortitude, with sheer obstinacy, often without information or communications or the knowledge of the whereabouts of friends. In less than a fortnight the enemy was stopped and the Americans were preparing to resume the offensive. The battle ground of Ardennes we may see in Figure 3.¹⁰ ([9]; [32]).

2.5.1 Weather and terrain analysis of Ardennes¹¹

Also spelled Ardennes, wooded plateau covering part of the ancient Forest of Ardennes, occupying most of the Belgian provinces of Luxembourg, Namur, and Liège;

part of the Grand Duchy of Luxembourg; and the French department of Ardennes. It is an old plateau comprising the western extension of the Middle Rhine Highlands, stretching in a northeast-southwest direction and covering more than 3,860 sq mi (10,000 sq km). Its geological history is complex; as a result of intense folding, faulting, uplifts, and denudations, some older strata of rock have been thrust over younger strata.

The name Ardennes used in a strict sense refers to the southern half of the area, where the elevations range from 1,150 to 1,640 ft [350 to 500 m], though the high point at Botrange, south of Liège, is 2,277 ft. This part consists of sandstone, quartzite, and some slate and limestone. Its rounded summits are separated by shallow depressions containing

10 http://www.army.mil/cmh-pg/brochures/ardennes/p04(map).jpg

11 http://ww2fighters.org/forums/index.php?showtopic=1112