A Dynamic Model for Winning Probability Estimation in a Long-Lasting Campaign

doi:10.23940/ijpe.18.12.p2.29152926

Abstract

Abstract:

Warfare is still unavoidable in the contemporary era, and it is essential for a country to estimate its winning probability when making decisions. While most previous studies are concerned with the result of a single battle, this paper focuses on a long-lasting campaign that may consist of many battles. First, this paper considers two parties and a single type of armed force. Moreover, a dynamic approach is proposed to describe the evolution of the campaign. In particular, each party has a certain number of armed forces in the beginning, and the campaign ends if all the armed forces for any party are extinguished. The probability for each party to win and the probability of a draw are analyzed. Numerical examples are presented to illustrate the applications.

Key words: warfare, winning probability, campaign, Markov process, armed force

Kaiye Gao, Xiangbin Yan, Rui Peng, Hui Qiu, and Langtao Wu. A Dynamic Model for Winning Probability Estimation in a Long-Lasting Campaign [J]. Int J Performability Eng, 2018, 14(12): 2915-2926.

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Figures/Tables 9

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References 15

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A	The first party
B	The second party
A	The number of armed forces that party A owns
B	The number of armed forces that party B owns
M	The amount of armed forces for party A in the beginning
N	The amount of armed forces for party B in the beginning
T	State transformation matrix in the case where the probabilities for events are cumulated
$\tilde{T}$	State transformation matrix in the case where the probabilities for events are not cumulated
Q(0)	Initial state distribution
Q(K)	State probability at the K^th day after the campaign beginning
$\tilde{Q}\left( K \right)$	State probability transiting from non-ending states on K^th day after the campaign beginning
PA(K)	Probability of A winning that happens during the time from day 1 until day K
PB(K)	Probability of B winning that happens during the time from day 1 until day K
PD(K)	Probability of a draw that happens during the time from day 1 until day K
PU(K)	Probability of an uncertainty that happens during the time from day 1 until day K
$\tilde{P}\tilde{A}\left( K \right)$	Probability of A winning that happens just on day K
$\tilde{P}\tilde{B}\left( K \right)$	Probability of B winning that happens just on day K
$\tilde{P}\tilde{D}\left( K \right)$	Probability of a draw that happens just on day K
$\tilde{P}\tilde{U}\left( K \right)$	Probability of an uncertainty that happens on day K
ET	Ending time
$\Pr (ET=K)$	Probability of ending at day K
ED	Expected duration

Day	A wins	B wins	Draw	Uncertain
1	0.09	0.09	0.02	0.8
2	0.187	0.187	0.046	0.58
3	0.2675	0.2675	0.07	0.395
4	0.3267	0.3267	0.089	0.2575
5	0.3674	0.3674	0.1027	0.1625
6	0.3939	0.3939	0.1121	0.1
7	0.4107	0.4107	0.1182	0.0603
8	0.421	0.421	0.1221	0.0358
9	0.4273	0.4273	0.1245	0.0209
10	0.431	0.431	0.126	0.0121
11	0.4331	0.4331	0.1268	0.0069
12	0.4344	0.4344	0.1273	0.0039
13	0.4351	0.4351	0.1276	0.0022
14	0.4355	0.4355	0.1278	0.0012
15	0.4357	0.4357	0.1279	0.0007
16	0.4358	0.4358	0.1279	0.0004
17	0.4359	0.4359	0.128	0.0002
18	0.436	0.436	0.128	0.0001
19	0.436	0.436	0.128	0.00006

Day	A wins	B wins	Draw	Uncertain
1	0.3	0.02	0.03	0.65
2	0.48	0.06	0.06	0.4
3	0.585	0.095	0.0825	0.2375
4	0.645	0.12	0.0975	0.1375
5	0.6787	0.1362	0.1069	0.0781
6	0.6975	0.1462	0.1125	0.0437
7	0.7078	0.1522	0.1158	0.0242
8	0.7134	0.1556	0.1177	0.0133
9	0.7165	0.1576	0.1187	0.0072
10	0.7181	0.1587	0.1193	0.0039
11	0.7190	0.1593	0.1196	0.0021
12	0.7195	0.1596	0.1198	0.0011
13	0.7191	0.1598	0.1199	0.0006
14	0.7199	0.1599	0.1199	0.0003
15	0.7199	0.1599	0.1200	0.0002
16	0.7200	0.1600	0.1200	0.0001
17	0.7200	0.1600	0.1200	0.000005

Day	A wins	B wins	Draw	Uncertain
1	0.09	0.09	0.02	0.8
2	0.097	0.097	0.026	0.58
3	0.0805	0.0805	0.024	0.395
4	0.0593	0.0593	0.019	0.2575
5	0.0406	0.0406	0.0138	0.1625
6	0.0266	0.0266	0.0094	0.1
7	0.0168	0.0168	0.0061	0.0603
8	0.0103	0.0103	0.0039	0.0358
9	0.0062	0.0062	0.0024	0.0209
10	0.0037	0.0037	0.0014	0.0121
11	0.0022	0.0022	0.0009	0.0069
12	0.0012	0.0012	0.0005	0.0039
13	0.0007	0.0007	0.0003	0.0022
14	0.0004	0.0004	0.0002	0.0012
15	0.0002	0.0002	0.0952	0.0007
16	0.00001	0.00001	0.000005	0.0004
17	0.000007	0.000007	0.000003	0.0002
18	0.000004	0.000004	0.000002	0.0001
19	0.000002	0.000002	0.000001	0.00006

Day	A win	B win	Draw	Uncertain
1	0.3	0.02	0.03	0.65
2	0.18	0.04	0.03	0.4
3	0.105	0.035	0.0225	0.2375
4	0.06	0.025	0.015	0.1375
5	0.0338	0.0163	0.0094	0.0781
6	0.0187	0.01	0.0056	0.0437
7	0.0103	0.0059	0.0033	0.0242
8	0.0056	0.0034	0.0019	0.0133
9	0.003	0.002	0.0011	0.0072
10	0.0016	0.0011	0.0006	0.0039
11	0.0009	0.0006	0.0003	0.0021
12	0.0005	0.0003	0.0002	0.0011
13	0.000025	0.000018	0.000001	0.0006
14	0.00001	0.000009	0.000005	0.0003
15	0.000007	0.000005	0.000003	0.0002
16	0.000004	0.000003	0.000002	0.0001
17	0.000002	0.000001	0.0000008	0.000005