warlord games kapu aditya rao

Kapu Aditya Rao

Warlord Games: Devising a

New Metric to Explain

Kuomintang Power

by Kapu Aditya Rao

Prof. Hong Li

8/06/2014

Heritage of China

Kapu Aditya Rao

Table of Contents

Abstract page 3

Introduction page 3

Data page 6

Theoretical Framework page 8

Analysis page 10

Conclusion page 11

Bibliography page 13

Appendix Page 14

Kapu Aditya Rao

Abstract

Historical analyses of Kuomintang (KMT) successes in unifying China during the late

1920s are usually qualitative and/or vague in nature. This paper seeks to understand

whether or not coalition power was the cardinal reason the KMT were successful in their

Northern Expedition during the Warlord Era with mathematical rigor. The paper utilizes

the Banzhaf Power Index to determine KMT strength relative to the strength of other

warlords. Intuitively, warlords with greater power must have had a greater role in the

Northern Expedition. Yet most of the credit conventionally falls on the KMT alone. By

using n-person game theory, we hope to answer an important question via unorthodox

means: “Who was the most powerful warlord during the Northern Expedition?”

Introduction

The Warlord Era of China roughly stretches from 1916 to 1928.1 Though the warlords

were living the a modern era with railroads, telegrams and automobiles, the fragmented

political situation looked more like a series of feudal lords struggling for dominance

amidst a backdrop of technologically and militarily superior foreigners slowly capturing

as many concessions as possible. Assassinations, strikes, massacres, betrayals and

intrigue were political norms. It was an era of aspiring ideals and of brutal realities,

colored by student protests, roving bandits, and nationalist rhetoric. But despite the

chaos, certain norms and structures could be detected.

1 Bonavia (p. 2).

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Warlord cliques were generally small-coalition structures: a warlord at the head,

senior officers below, and the rank and file at the bottom. Warlords, though emerging

from the carcass of Republican China, were themselves certainly not democratic. Many

warlords were probably ex-bandits. Zhang Zuolin, the leader of the Fengtian (or

Manchurian) clique was one such warlord.2 Warlords had the ability to tax the populace,

collect various duties, and maintain a confederation of lesser warlords; they were not at

all different from feudal kings of the past. Large warlords, from time to time, would ally

with one another and form coalitions. However, only the Kuomintang comes to mind

when we must imagine a victor of the free-for-all. In a brutal world with shifting alliances

and high stakes, was the KMT really the most important nationalist unifier it is pictured

to be?

Historical explanations as to why Chiang Kai-shek and the Kuomintang were able

to succeed in their Northern Expedition are generally qualitative in nature. Analysis of

the Warlord Era has usually fallen under the aegis of historians, not political scientists.

This paper hopes to create a bridge between more qualitative approaches favored by

the historian and qualitative methods used by the political scientist. Consequently, this

paper is experimental in nature and may fall under the category of “Cleometrics”.

Specifically, the paper uses the Banzhaf Power Index (BPI) to calculate the edge

one warlord may have over others. The BPI measures power based on how vital a

certain player is. If a player is never vital (low BPI score), they are irrelevant. Extremely

relevant players (high BPI score), however, can have a say in what bills get passed,

who is nominated for a position, what president is elected, etc. Philip D. Straffin

2 Bonavia (p. 63).

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presents a history and description of the Index: John Banzhaf sued the Nassau County

Board in 1965 because he believed that voters from certain districts were

underrepresented compared to others: “Banzhaf reasoned that a voter only has a direct

effect on the voting outcome when he is a swing voter in some winning coalition, and

hence a voter’s power should be proportional to the number of coalitions in which that

voter is a swing voter.”3 A useful, abstract example of how the BPI is calculated is found

in Appendix 1. We assume each troop to count as a “vote”. If a certain player is a swing

voter in all possible winning coalitions more times than another, the former player is

considered to have a higher BPI score. We resort to the BPI method because the

outcome of the “Warlord Game” is non-obvious and this paper seeks a nonconventional

explanation: the KMT held only Guangdong, yet they seem to be considered to be a

major player in unifying China. It is easy to be skeptical in thinking that they did it alone.

But how much did they contribute to the Northern Expedition? How much did other

warlords? Those are the guiding questions of the analysis of this paper.

The Banzhaf Power Index is preferred over other indexes of power such as the

Shapley-Shubik Power Index because the BPI does not particularly concern itself with

when a warlord joins a coalition; order does not matter in our analysis. Using the BPI

has further implications, including realizations of whether or not superiority in numbers

is what truly matters in warfare or not.

3 Straffin (p. 184-185).

Kapu Aditya Rao

Data

Listed below are the major warlord generals and the number of troops in their following

in the year 1926 along with brief descriptions. The total number of troops mobilized at

the time was around 600,000.4 To make the BPI calculations easier, below each warlord

is given a letter from A-F.

Chiang Kai-shek (A) – Leader of the Kuomintang, the Nationalist Party, based primarily

in Guangdong Province. By 1926, the KMT army “conglomerate” possessed around

100,000 troops.5

Sun Chuanfang (B) – The Nanjing warlord had 200,000 troops—on paper. The reality

was that these were separated amongst several commanders who did not all heed his

call.6 Realistically, he had around 70,000 troops willing to actually fight.7

Yan Xishan (C) – The utopian warlord of Shanxi had around 100,000 troops at the time

of the Northern Expedition.8

Wu Peifu (D) – With only an estimated 30,000 troops, the central China warlord was

betrayed by Feng Yuxiang while assaulting the Fengtian Clique in Manchuria. With most

of his troops absorbed, he escaped with crack troops to Central China where he would

gather his strength only to bear the brunt of the KMT’s Northern Expedition. How do we

4 Ch’i (p. 138).5 Jordan (p. 263).6 Jordan (p. 82).7 Ch’I (p. 138).8 Jordan (p. 273).

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arrive at 30,000 troops? This number is an estimate calculated by simple arithmetic:

Ch’i reports that 600,000 troops were mobilized in 1926; by calculating the number of

troops for the other warlords, we can infer to how many Wu Peifu controlled.

Feng Yuxiang (E) – The Christian-moralist general was in an uneasy stalemate with the

Fengtian Clique over control of (Northern) China by 1925. By 1926 he had around

100,000 troops involved in combat in Hunan alone.9 If he was to be equally matched

with Zhang Zuolin,10 we must assume that he had at least 50,000 troops to defend the

other parts of his territory including far North into Inner Mongolia.

Zhang Zuolin (F) – The Manchurian warlord and leader of the Fengtian Clique mobilized

around 150,000 troops.11

Certain omissions are salient, but necessary to simplify the Warlord Game. Japan and

many other foreign powers are absent. This is for two reasons: foreign influence

(especially intangible aid such as training) are difficult to record or account for12;

secondly, nearly all warlords—and certainly the ones listed above—had some

connection or other with foreigners, be they Russian, Japanese, British or American.

For similar reasons, artillery, training and a host of other troop “modifying” variables

were omitted. This is because of the game theoretic assumption that if one warlord was

to have an advantage over another in terms of troop quality, artillery pieces, etc., then

9 Jordan (p. 271).10 Waldron (p. 228).11 Spence (p. 350).12 Econometric analyses, however, could use a dummy variable to represent foreign training.

Kapu Aditya Rao

the other warlords would scramble to match the former. The shaky peace that reigned

between coalition conflicts could be seen as the time for arms races.

The Warlord Era was not a just a competition amongst large military strongmen

placed into well-defined blocs. At a glance, the whole system seemed nearly feudal:

most sources consulted for this paper describe series of confederations interacting on a

massive scale. Any bandit with a small following could be considered a warlord. A

Banzhaf Power analysis could probably be undertaken within each warlord coalition,

and each sub-coalition! For simplicity’s sake, this paper’s model considers the major

warlord factions to be more or less monolithic: if a warlord demands troops to be moved

someplace, it is moved there. There are no bills on the floor for debate, no legitimate

petitions against laws, and the warlord at the top alone determines policy outcomes

from wars.

Theoretical Framework

We conduct our analysis within the framework of a “Warlord Game”. The Warlord Game

is a simultaneous-action game with n-number of players (warlords). To “win” the game,

as Chiang Kai-shek and the KMT and their allies did by 1928, a player or a coalition of

players must possess at least 51% of the troops in the nation. By possessing the

majority of troops, we assume that the warlord coalition becomes a winning coalition.

Even if a player is a member of the winning coalition, they are still considered to have

won. Warlords are not created equal. Some warlords possess more troops than others.

At least at the start of the game (1926), no single warlord is the dictator, a player who by

himself possesses 51% of the strength of the country. To make the math a bit easier,

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we consider all troop numbers to be in terms of 1,000 soldiers. So, a warlord with

100,000 soldiers will have 100 votes, a warlord with 150,000 soldiers will have 150

votes, etc.

We now solve the game:

1. [(0.51)x(total # of troops in China); (KMT troops), (Sun Chuanfang’s troops), (Yan

Xishan’s troops), (Wu Peifu’s troops), (Feng Yuxiang’s troops), (Zhang Zuolin’s

troops)]

2. [q; A, B, C, D, E, F] where q is the number of votes/troops required to win the

game

3. [306; 100, 70, 100, 30, 150, 150]

4. To find KMT power, we write down possible winning coalitions and mark the

number of opportunities for swing voting using a computer program13

5. The power of each voter as a percentage:

Number of votes Power

150 25.86 %

100 15.52 %

70 15.52 %

30 1.72 %

13 Temple University provides a free, online version at http://cow.math.temple.edu/~cow/cgi-bin/manager

Kapu Aditya Rao

6. Therefore:

Warlord Number of Troops BPI

KMT 100,000 15.52 %

Sun Chuanfang 70,000 15.52 %

Yan Xishan 100,000 15.52 %

Wu Peifu 30,000 1.72 %

Feng Yuxiang 150,000 25.86 %

Zhang Zuolin 150,000 25.86 %

Analysis

We see that the true power-brokers of the time appeared to be Feng Yuxiang and

Zhang Zuolin. Wu Peifu is hopelessly weak--in only 1.72% of all possible winning

coalitions does he have the swing vote. The KMT, Sun Chuanfang and Yan Xishan are

equally powerful. We cannot say that in the year 1926, the KMT was posed to win

because it had a higher BPI than the other warlords. That honor falls to Feng Yuxiang

and Zhang Zuolin.

By the end of the Northern Expedition, Chiang Kai-shek was not the sole ruler of

China. Feng Yuxiang and Yan Xishan would continue to clash with him in the future,

especially during the Central Plains War in 1930. KMT victory in 1928 differed from KMT

victory in 1930. The Northern Expedition Warlord Game (1926-1928) was different from

Kapu Aditya Rao

the Central Plains War Game (1930). In 1926, Chiang Kai-shek faced several

opponents and thus had to form a coalition with Yan Xishan and Feng Yuxiang. In 1930,

Chiang Kai-shek was probably powerful enough to challenge them and emerge

victorious. It is therefore probable that Feng Yuxiang believed that, with Zhang Zuolin

gone by 1929, he was and would continue to be the main power-broker in China. With

defeat in the Central Plains War, he was proven wrong.14

The analysis shows us that minority powers are still capable of being coalition

formers. However, minorities, if they stay minorities, will not be able to dictate victory

terms as well as more powerful players.

The BPI can be used to “rank” players based on how decisive their participation

was to victory. The KMT, contrary to conventional belief, was not the star player for the

Northern Expedition, though the party created the concept in the first place. It is, in fact,

Feng Yuxiang, whose aid allowed for the Expedition to achieve success.

Conclusion

This paper began with the assumption that the KMT alone emerged victorious in 1928.

However, as the analysis shows, the KMT did not—could not—emerge victorious alone

by 1928. Instead, it was a three-man coalition between Chiang Kaishek, Yan Xishan,

and Feng Yuxiang. This paper stops short at 1928. But if we were to extend the game

into the Central Plains War of 1930 and consider it to be a two-part, turn-based game,

then the result would be worthy of much debate.

14 Bonavia (p. 115).

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The Banzhaf Power Index ultimately serves as a mechanism not to gauge who

will win (for could it not have simply been Zhang Zuolin and Feng Yuxiang in a two-man

coalition?), but rather what is the probability of a player changing the outcome of a

game by changing his/her votes. This is important because the higher a player’s ability

to change the winner of the game, the better that player can dictate the terms of victory.

By 1929, Feng Yuxiang, Chiang Kai-shek and Yan Xishan were able to decide a very

important victory term: that they would still remain in power. But after the Northern

Expedition, relative power changed, with the KMT somehow possessing more than the

other extent warlords.

Future studies on this matter may take different approaches. Primarily, the

Warlord Game could be seen as a sequenced game with several rounds, each round

having a different outcome for the next one, rather than the simultaneous action game

used in this paper. This way, a KMT victory in 1926 could change things for 1927.

Spatial models may also be effective. Sun Chuanfang and Wu Peifu, both weaker than

the KMT, were the first to bear the brunt of their advance. After their troops were

defeated and absorbed, the scales of power and the BPI tips in favor of the KMT.

Studies that use a BPI model that allows for the elimination of non-winning coalition

members would certainly benefit from added clarity.

Kapu Aditya Rao

Bibliography

Bonavia, David. China's Warlords. No ed. Hong Kong: Oxford University Press, 1995.

Chi, Hsi. "Military Capabilities: Weaponry and Tactics." In Warlord Politics in China, 1916-

1928. Stanford, Calif.: Stanford University Press, 1976.

Jordan, Donald A. The Northern Expedition: China's National Revolution of 1926-1928. No

ed. Honolulu: University Press of Hawaii, 1976.

Lary, Diana. Warlord Soldiers: Chinese Common Soldiers, 1911-1937. Cambridge

[Cambridgeshire: Cambridge University Press, 1985.

Spence, Jonathan D. "Chapter 4: The Clash." In The Search for Modern China. No ed. New

York: Norton, 1990.

Straffin, Philip D. "Application to Politics: The Banzhaf Index and the Canadian Constitution."

In Game Theory and Strategy. No ed. Washington: Mathematical Association of America, 1993.

Waldron, Arthur. From War to Nationalism: China's Turning Point, 1924-1925. No ed.

Cambridge: Cambridge University Press, 1995.

http://cow.math.temple.edu/~cow/cgi-bin/manager (computer program used).

http://cow.math.temple.edu/~cow/cgi-bin/manager

Kapu Aditya Rao

Appendix I

This paper is an exercise in the field of Game Theory, specifically N-Person Game

Theory. Of this sub-field, the Banzhaf Power Index is merely one model out of many.

The following example is taken from Philip D. Straffin’s (1993) excellent book on

game theory, Game Theory and Strategy. Suppose in some hypothetical country there

are three voters in a legislative body. In order to pass a bill, a simple majority is needed.

Furthermore, there are only four votes in the system; but voters are all given different

number of votes. Therefore, some voters appear to be more powerful than others. In our

example, we will presume that Voter A has the ability to cast two votes for or against a

bill, but Voters B and C only have one vote each. Subsequently, three votes (q) are

required to pass a bill.

[q; w1, w2, w3]

[3; 2, 1, 1]

A, B, C

Now we calculate all possible winning coalitions. Then, in the same way Straffin

does, we will underline voters whose defections would cause the bill to fail:

AB, AC, ABC

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In the winning coalition AB, if A leaves the coalition, their bill does not pass;

likewise, the bill fails if B leaves too. We then take the number of swing possibilities and

divide by the total number of possible swings:

So, for voter A, we determine:

Voter A can threaten to break up a coalition three times. Voter B and C, however,

can only do so in one instance separately.

Voter Name Banzhaf Power

Voter A = 0.6 = 60%

Voter B = 0.2 = 20%

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Voter C = 0.2 = 20%

We see here that though Voter A possesses two the number of votes than either

Voter B or Voter C, his power is disproportionately larger. Based on the BPI, he is thrice

—not twice—as powerful in this game. Voter A is not the dictator of the game. The

dictator is a voter who possesses q votes all by him or herself. In the game above, a

voter would be a dictator if he or she possessed six votes. This voter would therefore

have no need to create a coalition of any sort and can pass a bill at will.

In the Warlord Game, we substitute troops for votes and warlords for voters.

There are a few implications that follow. First, if a voter has 0% Banzhaf Power,

then he or she is irrelevant. This is the scenario that John Banzhaf confronted for the

Nassau County Board. If we had a voter that was not crucial to whether or not a bill

passed regardless of the coalition, then they might as well not even take part in the

game! Second, players who start with fewer votes are not necessarily utterly powerless.

They may still be necessary to take some part in winning the game. Higher BPI

percentages reflect the ability of a voter to be a more important part of possible

coalitions. We conjecture that the more powerful a voter is, the more the bill in question

will benefit him or her. This is especially relevant when voters are not attempting to pass

a bill, but rather elect a leader. The bulk of this paper essentially sought to answer the

question of whether or not Chiang Kai-shek emerged in the leadership position by the

end of the 1920s.

Kapu Aditya Rao

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