The Salvo Combat Model with a Sequential Exchange of Fire 1

Transcription

1 Abstract The Salo Combat Model with a Seqential Exchange of Fire 1 Michael J Armstrong, Associate Professor Goodman School of Bsiness, Brock Uniersity, St Catharines, ON, LS 3A1, Canada michael.armstrong@brock.ca This paper deelops a ersion of the stochastic salo combat model in which the exchange of fire is seqential, rather than simltaneos. This seqential-fire ersion is bilt by modifying the eqations in the original simltaneos-fire ersion. The performance of the seqential model is tested by comparing its otpts to those of a Monte Carlo simlation. The fit between the model and the simlation is ery close, especially for the mean and standard deiation of losses. The model is then applied to the Battle of the Coral Sea. The reslts sggest that attacking first wold hae gien the American force a larger adantage than that proided by an extra aircraft carrier. Keywords: Military, Defence, Simlation, Naal tactics, Salo eqations, Stochastic model. Introdction One way for a military force to seek an adantage oer its opponent is to attack first. Historically, naies hae achieed first strikes ia strategic srprise (Pearl Harbor in 1941), sperior scoting (Midway in 194), tactical positioning (Philippine Sea in 1944), or longerranged weapons (Latakia in 1973). The potential benefits of striking first, and the potential costs of sffering sch an attack, are of on-going interest to naal analysts (see, e.g., Lcas and McGnnigle, 003; Tiah, 007; Edmiston, 011). It een has been arged that attacking effectiely first shold be a primary goal of naal tactics (Hghes, 000: xx). The possibility of striking first has also been stdied in other types of conflict, sch as terrorist attacks. For example, some researchers hae analysed game-theory models where terrorists choose which targets to attack, while goernment forces choose which targets to defend; see, for example, Zhang and Bier (007), He and Zhang (01), or Hasken and Zhang (01). In some of these attacker-defender games the attacking terrorists moe first, while in other cases they moe at the same time or after the defending goernment forces. Gien the importance of this factor in warfare past and present, it wold be sefl for analysts and historians to hae a simple way to model missile combat where the two sides attack seqentially, instead of simltaneosly. The salo eqations of Hghes (1995) already proide a simple way to represent combat inoling missiles or airstrikes, mch as the Lanchester models do for gnfire combat (see, e.g., MacKay, 009; Johnson and MacKay, 011; Atkinson et al, 1 This is a post-peer-reiew, pre-copyedit ersion of an article pblished in the Jornal of the Operational Research Society. The definitie pblisher-athenticated ersion [Armstrong MJ, 014. The salo combat model with a seqential exchange of fire, Jornal of the Operational Research Society 65 #10, , doi: /jors ] is aailable online at: 1

2 01). Howeer, both the original deterministic salo model and the later stochastic one (Armstrong, 005) assme that the exchange of fire is simltaneos. This paper modifies the salo combat model to handle a seqential exchange of fire. In this ersion, one force exectes its attack first while the opponent defends; sbseqently, the opponent s sriors (if any) retrn fire while the original attacker defends. The research begins with the deterministic salo model (Hghes, 1995). Since this model is easily modified to handle seqential attacks, it allows an initial assessment of the ale of attacking first, and of how that ale is inflenced by the combat nits characteristics. The stdy then deries a seqential ersion of the stochastic salo model (Armstrong, 005). Since the salo model is a highly simplified representation of combat, the modifications are chosen to maintain that simplicity as mch as possible. The seqential ersion is tested by comparing its reslts to those of a Monte Carlo simlation, as in Armstrong (011). The nmerical reslts show that the model otpts fit ery closely with the simlation otpts, especially for smmary statistics sch as the mean and ariance of the losses. The last part of the paper applies the seqential stochastic model to the 194 Battle of the Coral Sea between American and Japanese aircraft carriers, as in Armstrong and Powell (005). This conterfactal analysis sggests that the Americans cold potentially hae gained a large adantage if they had been able to attack before the Japanese, rather than at the same time. This first-strike adantage wold hae been more alable than haing an additional aircraft carrier. Conersely, being forced to attack second wold likely hae been fatal to the American task force and reslted in the loss of Port Moresby to the Japanese. The Basic Salo Combat Model Consider a battle between two forces, Red and Ble. Let A represent the nmber of combat nits (e.g., warships) in the Red force at the beginning of the battle. Each one has offensie firepower α, which is the nmber of missiles (or airstrikes, etc.) accrately fired per salo at the enemy. Ble has B nits, each with defensie firepower z, which is the nmber of incoming enemy missiles intercepted per salo by actie defences. Each missile that is not intercepted cases the loss of portion of a Ble ship, so that the staying power x = 1/ is the nmber of hits needed to incapacitate one Ble ship. Similar symbols represent Red s defensie firepower y, staying power w, and loss per hit, along with Ble s offensie firepower β. See Table 1 for a smmary of this notation. The battle consists of Red firing a salo of missiles that Ble tries to intercept, while Ble simltaneosly lanches a salo that Red tries to intercept. Hghes (1995) deeloped the salo eqations to calclate the change in strength ΔB for Ble and ΔA for Red; the particlar notation shown below is that of Armstrong (005). ΔA = -(βb - ya) sbject to 0 -ΔA A (1) ΔB = -(αa - zb) sbject to 0 -ΔB B ()

3 For example, Eqation 1 first calclates Ble s total nmber of offensie missiles fired as βb, and then sbtracts Red s total nmber of defensie interceptions as ya. If this difference is zero or less, i.e., if (βb - ya) 0, this indicates that all of the missiles hae been intercepted, and Red sffers no loss; hence ΔA = 0. If the difference instead is positie, (βb - ya) > 0, this indicates the nmber of missiles that srie interception and hit Red ships; Red sffers loss for each sch hit. Of corse Red cannot lose more ships than it actally has, so its loss is capped at -ΔA = A. The interested reader shold see Hghes (1995) for frther details abot these eqations, or Armstrong (004) for an analysis of their mathematical properties. A Seqential Deterministic Model It is straightforward to modify the deterministic ersion (Hghes, 1995) of the salo model to depict seqential fire. Seeral nmerical stdies hae already sed it that way (Lcas and McGnnigle, 003; Tiah, 007; Edmiston, 011). Sppose that Ble attacks first while Red defends, and then the Red sriors (if any) retrn fire while Ble defends. With this seqence, Red sffers losses according to Eqation 1 and is left with A1 sriors as follows. A1 = A + ΔA = A - (βb - ya) = A - βb + ya sbject to 0 A1 A (3) If (βb - ya) A, then A1 = 0 and there is no Red retrn fire. Aside from that triial case, the change in Ble s strength ΔB and its sriing force B1 can be calclated as follows: (a) If α(a - βb + ya) zb, then Red s retrn fire is all intercepted; ths ΔB = 0 and B1 = B. (b) If α(a - βb + ya) B(x + z), or eqialently (αy + α)/(αβ + z + x) B/A, then Red s retrn fire completely eliminates Ble; ths ΔB = -B and B1 = 0. (c) Otherwise, Red s retrn fire eliminates part of Ble s force according to Eqation 4. ΔB = -(αa1 - zb) = -(α(a - βb + ya) - zb) = -(αa - αβb + αya - zb) (4) For example, sppose that Ble initially has B = 4 ships while Red has A = 5 ships. Let all ships be identical with offensie firepower α = β = 4 missiles per salo, defensie firepower y = z = interceptions per salo, and loss per hit of = = 1/3 ships. Ble s attack will eliminate (4x4 x5)/3 = ships, leaing 5 = 3 Red sriors. Those sriors will then retrn fire and case Ble to lose (4x5 4x 4x4x4/3 + 4xx5/3)/3 = 1.33 ships. If Red cold instead attack simltaneosly, then Ble s losses wold be (4x5 x4)/3 = 4 ships. Ths in this nmerical example, shooting first greatly aids Ble by redcing its losses, while shooting second leaes Red relatiely worse off for that same reason. Not srprisingly, the desirability of shooting first and the corresponding ndesirability of shooting second hold more generally in the salo model. To be precise, shooting first is always at least as good as shooting simltaneosly; and shooting simltaneosly is always at least as good as shooting second. Howeer, there are some special cases where different attack seqences proide the same otcomes, and ths are eqally desirable. 3

4 One sch case occrs wheneer either side is too weak to penetrate their opponent s defences; i.e., if (βb - ya) 0 or (αa - zb) 0. In this case, the weaker side sffers the same loss regardless of the attack seqence, while the stronger side sffers no loss regardless of the attack seqence. For instance, this will always be tre in low lethality battles (defined as haing /z y/), becase then at least one side is nable to hit the other (Armstrong, 004). It can also be tre in some high lethality battles (where /(z+x) (y+w)/) and moderate lethality battles (where /z > y/ bt /(z+x) < (y+w)/), if one side otnmbers the other by sch a wide margin that its defensie firepower can intercept all incoming fire. Another partial special case can occr in moderate or high lethality battles if one side s offensie firepower is sfficient to eliminate its opponent een when shooting second. For example, this wold apply for Red if α(a - βb + ya) B(x + z). Red wold then prefer to attack first and thereby aoid any losses itself; bt it wold be indifferent between attacking simltaneosly and attacking second, becase ΔB = -B and ΔA = -(βb - ya) either way. Conersely, Ble wold want to aoid attacking second, becase it wold be eliminated before it cold fire; bt it wold be indifferent between attacking simltaneosly and attacking first. Retrning to the preios nmerical example, sppose that Red now has A = 8 warships. Their defence cold completely absorb any attack from Ble, as (4x4 x8)/3 = 0, and their offence wold eliminate all the Ble ships. The otcome therefore wold be the same regardless of who attacks first, making Red and Ble indifferent abot the seqence. In sitations where none of these special cases holds, it can be informatie to compare Ble s losses when firing first to those it sffers when firing simltaneosly or second. ΔB1st / ΔBOther = ( -(αa - zb - αβb + αya) ) / ( -(αa - zb) ) (5) = 1 - (α(βb - ya))/(w(αa - zb)) The ratio on the right of this expression represents the relatie redction in Ble s losses de to attacking first. It can be shown that this benefit is increasing in Ble s strength B, offensie firepower β, and defensie firepower z. Conersely, the adantage is decreasing in Red s strength A, offense α, defence y, and staying power w. For example, the effect of Red s offensie fire is fond by taking the deriatie of the ratio with respect to α. / α (α(βb - ya)) / (w(αa - zb)) (6) = ( (βb - ya)w(αa - zb) (α(βb - ya))(wa) ) / (w (αa - zb) ) = ( w(βb - ya)(αa - zb αa) ) / (w (αa - zb) ) = - ( zb(βb - ya) ) / (w(αa - zb) ) This deriatie is negatie when (βb - ya) > 0 and (αa - zb) > 0, i.e., each side can hit the other. Another way to assess the effect of attacking seqentially rather than simltaneosly is ia the fractional exchange ratio (FER). This ratio compares the fraction of Ble forces lost to the fraction of Red forces lost, as in FER = (ΔB/B) / (ΔA/A). FER = 1 indicates that both sides are losing the same proportion of their forces, so that the battle is a tie in some sense. The 4

5 force ratio B/A where this happens is what Armstrong (004) called the parity point, and it proides one measre of the relatie combat worth of the warships on each side. To see this, consider again the preios nmerical example. The Red and Ble ships hae identical characteristics, so when attacking simltaneosly they are clearly of eqal ale. FER = 1 wheneer both sides hae the same nmber of ships, i.e., wheneer B/A = 1. Howeer, if Ble attacks first instead of simltaneosly, then soling Eqations 3 and 4 shows that FER = 1 only when B/A = That is, each Red ship firing second is degraded to the eqialent of only Ble ships firing first; or conersely, each first-firing ship becomes the eqal of 1/0.573 = 1.74 second-firing ships. This particlar nmber is scenario-specific, bt it proides a clear illstration of the force mltiplier benefit of striking first. A Seqential Stochastic Model When modifying the stochastic ersion of the salo model for seqential attacks, the main challenge is that the nmber of sriors who retrn fire is a random ariable. This section describes one relatiely straightforward way to handle that complication. In the original stochastic salo model (Armstrong, 005), the nmber of accrate missiles αi fired by each Red ship is treated as an independent and identically distribted random ariable that follows a binomial distribtion with mean = n p and ariance σ = n p(1- p). Here, n represents the nmber of missiles fired per ship, and p represents each one s probability of sccess. The nmber of interceptions zj by each Ble ship is likewise assmed to be binomial with parameters z = nz pz and σz = nz pz(1- pz). The total nmber of Ble s defensie interceptions DefB is sbtracted from the total nmber of Red s offensie missiles OffB to get the net nmber of non-intercepted missiles NetAB = OffA - DefB. This is assmed to follow a normal distribtion with cmlatie distribtion fnction (cdf) FNetAB and probability density fnction (pdf) fnetab. Each of these missiles hits a Ble ship and cases a random amont of damage k with mean and ariance σ. The nominal nmber of Ble sriors (i.e., before trncating the distribtion s tails to accont for 0 -ΔB B) follows a normal distribtion with cdf G and pdf g. Similar symbols represent Ble s attack and Red s defence. Table contains a smmary of this notation. Sppose now that Ble attacks first. Eqations 7 and 9 in Armstrong (005) proide expressions for the mean μa1 and ariance σa1 of the nmber of Red sriors A1; they are reprodced below. (The ± / terms represent a continity correction.) The interested reader shold refer to that paper for the deriation of these eqations. A1 F A / F 0 / f A / f 0 / A 1 F ( A / ) (7) 5

6 A1 F ( A / ) F (0 / ) A 1 F ( A / ) EA A / f ( A / ) 0 / f (0 / ) In the seqential model, Red s retrn fire OffA1 consists of the total nmber of missiles fired by their sriors. It therefore is the sm of a random nmber A1 of random ariables αi, with the following mean and ariance (see, e.g., Ross, 1993: 94-98). E[OffA1] = μa1 (9) Var[OffA1] = μa1σ + μ σa1 (10) Eqation 9 is similar to the corresponding one in Armstrong (005). In Eqation 10, the second term reflects the added ariation de to the randomness of Red s strength A1. Eqations 9 and 10 for Red s attack are combined with existing ones from Armstrong (005) for Ble s defence to get the nominal nmber of non-intercepted offensie missiles NetAB. E[NetAB] = E[OffA1] - E[DefB] = μa1n p - Bnz pz (11) Var[NetAB] = Var[OffA1] + Var[DefB] = μa1n p(1- p) + n p σa1 + Bnz pz(1- pz) (1) Smming the random loss per missile oer the random nmber of missiles NetAB leads to the following expressions for the nominal nmber of Ble sriors. These replace Eqations 4 and 5 of Armstrong (005). = B - E[NetAB] E[] = B - (μa1 - Bnz pz) (13) = (μa1n p - Bnz pz) + (μa1n p(1- p) + Bnz pz(1- pz) + A1 n p ) (14) - (μa1n p - Bnz pz)gnetab(0) + (μa1n p(1- p) + Bnz pz(1- pz) + A1 n p )gnetab(0) These expressions can be sed to find the probability of arios otcomes for Ble, sch as the probability of no loss or of no sriors. They also lead to the mean and ariance of the actal nmber of Ble sriors B1, i.e., after acconting for 0 B1 B. P P B B 1 G B *( B / ) (15) 1 1 B 0 *(0 / ) 1 G B 1 (16) 1 (8) E B G B / G 0 / g B / g 0 / 1 B 1 G ( B / ) (17) 6

7 Var B G ( B / ) G (0 / ) 1 B 1 G ( B / ) EB 1 (18) B / g ( B / ) 0 / g (0 / This approach reqires relatiely few changes to the original simltaneos-fire eqations and retains their ease of software implementation. Howeer, it cold be arged that it is oerly simplistic, as the only information it ses abot the nmber of Red sriors is the mean and ariance. The shape of A1 s distribtion is ignored, inclding the trncation of its tails at 0 A1 A. A more sophisticated approach cold explicitly accont for this trncation: e.g., if A1 = 0, then there is no retrn fire; or if A1 = A, then eery Red nit retrns fire. Fortnately, the nmerical testing in the next section sggests that sch added complexity is nnecessary. In this seqential stochastic model, the same preferences for attack seqence apply as in the deterministic case, bt with two changes. Firstly, the preferences now need to be expressed in terms of aerages or probabilities: e.g., shooting first is at least as good on aerage as shooting simltaneosly, and shooting simltaneosly is at least as good on aerage as shooting second. Bt in any gien instance, a force that shoots second and is lcky cold achiee better reslts than an identical force that shoots first bt is nlcky. Secondly, the bondary cases for this seqence preference are gradal tendencies rather than distinct ct-offs. According to the deterministic model, a fleet that is mch stronger or weaker than its opponent shold be indifferent to the seqence of attacks, as the battle s otcome wold be the same in any eent. In the stochastic model, howeer, there is always a small chance that an inferior force cold penetrate its opponent s defences with at least one missile. This sggests that een a greatly sperior force shold prefer to attack first, particlarly if the leaders of the nay or its goernment are relatiely risk-aerse. Testing the Model with Monte Carlo Simlation This section compares the performance of the seqential-fire stochastic salo model to that of a Monte Carlo simlation, mch as Armstrong (011) did with the simltaneos-fire ersion. The seqential salo model (hereafter, the model ) is implemented in Excel spread sheet software, as in Armstrong (007, 011). The Monte Carlo simlation ( the simlation ) is also set p in Excel, sing software add-in. The same parameter inpts (scenarios) are sed in both cases, and then their otpts (battle otcomes) are compared to see how closely they match. (The spread sheets are aailable from the athor pon reqest.) Data Generation The inpt parameters for this nmerical stdy are as follows. The size B of the Ble force (which shoots first) is fixed at 4 warships, while the size A of the Red force (which shoots second) is aried from to 10 warships. These choices proide scenarios where the likely Ble ) 7

8 otcome ranges from almost always being nharmed (when A = ), to almost always being eliminated (when A = 10). For simplicity, all Red and Ble ships are identical: each can lanch n = nβ = 4 offensie missiles and attempt ny = nz = defensie interceptions per salo. The probability of sccess for each offensie missile p = pβ is aried across 3 ales: 0.50, 0.67, and The same 3 ales are separately sed for the probability of sccess for each defensie interception py = pz. The damage cased by each missile is random with mean μ = μ = 0.33 ships and standard deiation σ = σ = 0.11 ships; ths the aerage staying power is 1/ hits per ship. These are the same figres sed in Armstrong (007, 011). The combination of 9 Red forces sizes, 3 offensie missile probabilities, and 3 defensie interception probabilities gies a total of 81 scenarios for testing in both the model and the simlation. In terms of salo combat lethality (Armstrong, 004), these scenarios all fall into the moderate lethality category. The stdy aoids high lethality scenarios becase their tendency towards heay losses or oerkill cold oershadow any differences between the model and the simlation. The stdy also aoids low lethality scenarios, becase if one side is effectiely inlnerable, it does not matter whether it fires first or second. For the simlation the first half of the spread sheet calclates the losses inflicted by Ble against Red. A binomial distribtion represents the nmber of accrate missiles from Ble, and another binomial distribtion represents the nmber of sccessfl interceptions by Red. The difference (if positie) between those two ariables indicates the nmber of non-intercepted missiles that hit Red. For each missile, a normal distribtion determines the loss sffered by Red. The second half of the spread sheet calclates the losses inflicted by the Red sriors (if any) against Ble in a similar manner. The simlation tracks the Ble losses across trials (battles) for each of the 81 scenarios. Each scenario yields for metrics: the mean losses sffered by Ble, measred in ships; the standard deiation of those losses, in ships; the probability of losing zero ships; and the probability of losing all 4 ships. The mean and standard deiation are key smmary statistics of the loss distribtion, while the two probabilities examine the distribtion s pper and lower tails. Data Analysis As an example of the reslts, consider the scenario where Red has 6 ships and the probability of sccess is 0.67 for each offensie missile and defensie interception. Across simlated trials, Ble losses had a mean of.637 ships and a standard deiation of ships. Ble lost no ships in 0.8% of the trials, while it lost all of its ships in 1.5% of them. By comparison, the model calclates Ble s mean loss to be.643 ships, or ships higher than the simlation. The standard deiation is ships, or ships lower. The probability of losing zero ships is 1.1% and the probability of total loss is 15.%. Ths in this particlar scenario, the fit between the model and the simlation is ery close for the mean and standard deiation, and reasonably close for the tail probabilities. 8

9 The next step is to assess the fit between the model and the simlation across all 81 scenarios at once. One qalitatie approach ses scatter plots, as in Armstrong (011). For example, in Figre 1 each circle represents one scenario; it matches the probability of Ble being totally eliminated as calclated by the model (ertical axis) with the corresponding probability from the simlation (horizontal axis). The diagram also incldes a diagonal reference line to indicate where the dots hypothetically wold fall if a perfect fit existed between model and simlation. In this figre most points fall jst aboe the line, indicating that the model tends to slightly oerestimate this probability, especially for ales near 0.5. This may be de to the simple continity correction Armstrong (005) sed in the model, which assigns probability mass near zero to zero. Scatter plots for the other 3 metrics (not shown) sggest een closer fits. To confirm these isal impressions, linear regressions were calclated next for each of the 4 metrics across all 81 pairs of measrements, with the regression constants fixed to zero. The top half of Table 3 displays the reslts. In this context, slope coefficients and R ales close to indicate that the model otpts are similar to the simlation otpts. For example, regression of the probabilities of complete elimination from the model against those of the simlation yielded R = 0.999, showing that the model gies extremely close estimates of the simlation ales; it explains 99.9% of the ariation in them. The slope coefficient of 1.06 meanwhile confirms that the model tends to marginally oerestimate these probabilities. The reslts for the other 3 metrics indicate fits that are at least as good. The bottom half of the table qantifies the model-to-simlation fit in a different way. First, the difference between each reslt from the model and the eqialent reslt from the simlation was calclated, as in, e.g., (difference in mean loss) = (model mean loss) (simlation mean loss). Then the aerages and ariances of those differences across all 81 scenarios were determined. Aerage differences close to zero indicate that the model gies an accrate (nbiased) estimate of the simlation, whereas positie aerages indicate a tendency to oerestimate and negatie aerages a tendency to nderestimate. Similarly, ariances close to zero sggest that the model is consistent in its estimates of the simlation, whereas larger ariances sggest wider swings from scenario to scenario. The table also shows the difference with the largest magnitde, positie or negatie. All of these figres confirm that the fit between the model and the simlation is generally ery close. Applying the Model to a Carrier Battle This section demonstrates the application of the seqential stochastic salo model in a small stdy of the Battle of the Coral Sea (the Monte Carlo simlation is not sed in this part of the stdy). This battle was important historically becase it was the first naal engagement foght entirely ia an exchange of carrier airstrikes, and strategically becase it saed Port Moresby from Japanese inasion. At the tactical leel, the losses were heaier on the American side: the Lexington sank and the Yorktown needed major repairs in Pearl Harbor. On the Japanese side, the crippled Shokak reqired months to repair; the Zikak remained operational, bt retrned to Japan to replenish its aircraft (Lndstrom, 1984). 9

10 In the battle on 8 May 194, the attacks between the aircraft carriers (CVs) of the United States Nay (USN) and the Imperial Japanese Nay (IJN) were effectiely simltaneos. Bt they cold easily hae been seqential if, as at Midway, one fleet had spotted the other sooner. This work extends an earlier conterfactal stdy by Armstrong and Powell (005). They aried the nmber of USN CVs, the disposition of those CVs (groped or dispersed), and their ratio of defensie fighters to offensie bombers. Their reslts indicate that adding another USN CV wold hae sbstantially increased IJN losses, bt only marginally decreased USN losses. As well, the USN s otcome wold likely hae been better if it had dispersed its CVs, bt worse if it had carried more fighters and fewer bombers. The new factor to ary here is the seqence of attacks; three possibilities are compared. Simltaneos: The USN and IJN condct their airstrikes simltaneosly, as in the actal battle, and as in Armstrong and Powell (005). USN first: The USN exectes its attack first while the IJN defends; then the IJN sriors respond with a conter-attack while the USN defends. USN second: The IJN exectes its attack first while the USN defends; then the USN sriors respond with a conter-attack while the USN defends. The other parameter that is aried here is the nmber of American carriers. Historically the USN had CVs in the battle, bt this stdy also tries ale of 1, 3, and 4, as there were 4 USN CVs in the Pacific theatre at that time. Combining the 4 leels of USN force strength along with the 3 attack seqences yields 1 different scenarios for comparison. The model inpts for these scenarios are shown in Table 4. They were deried in Armstrong and Powell (005) sing historical data. Each point of offensie firepower represents one die bomber or torpedo bomber, and each point of defensie firepower represents one fighter aircraft. For example, each USN CV carries 17 fighters, and each of those fighters has a probability of of sccessflly intercepting one IJN bomber. Table 5 contains the main model otpts for both the USN and IJN, ronded to two decimal places. These indicate the aerage CV losses, the standard deiation of those losses, the probability of losing zero CVs, and the probability of losing all of the CVs. The final colmn sbtracts the mean USN loss from the mean IJN loss; a positie nmber indicates an American adantage, while a negatie one indicates a Japanese adantage. For example, the forth row of the table describes the historical case where the USN has CVs and attacks simltaneosly with the IJN; these nmbers are the same as in Armstrong and Powell (005). The model estimates the aerage nmber of CVs incapacitated (not necessarily snk) at 1.8 for the USN and 1.44 for the IJN. These are reasonably similar to, thogh perhaps slightly higher than, the actal reslts of the battle. The difference of the means, = CVs, can be seen as fairly representing the tactical defeat sffered by the USN. By comparison, the fifth row of Table 5 shows the reslts of the same USN force attacking first. The USN s aerage loss decreases from 1.8 to jst 0.17 CVs, while the 10

11 standard deiation remains relatiely nchanged at 0.18 erss 0.19 CVs. The probability of the American force sffering no loss at all jmps from 0% to 38.%, while the probability of being completely destroyed drops from 30.6% to 0%. The IJN loss statistics remain nchanged, so the difference in the aerage losses changes from (a small American disadantage) to +1.7 (a large American adantage). Looking down the rest of the table, the nmbers show that attacking first greatly redces the USN s losses wheneer it employs or more CVs. In this context, striking first is more alable than adding another CV. This point is illstrated in Figre, which shows the difference in mean losses from the model for each combination of carrier strength and attack type. While attacking first almost always proides some adantage to the USN, there is clearly an interaction effect between the seqence of attacks and the nmber of carriers. The incremental benefit of haing an extra CV is greatest when the USN attack firsts; or to pt it another way, the benefit of a first strike is greatest when there are enogh CVs to delier a decisie blow. By contrast, a lone USN CV receies little benefit from attacking first, as it lacks enogh firepower to fight opponents. To attack effectiely first (Hghes, 000: xx), one mst not only attack first, bt also attack effectiely. The plot also illstrates that more is not always better. If the USN attacks simltaneosly or second, they actally get better aerage reslts with 1 CV than with, becase the latter choice exposes more of their ships to loss. These reslts sggest that if the USN had actally been able to attack first with its CVs, the battle s tactical otcome wold likely hae been mch more faorable. The Lexington wold not hae snk and the Yorktown wold hae been ntoched. This wold hae gien the USN an extra margin of safety for the Battle of Midway one month later. Conersely, attacking second wold likely hae been disastros for the USN. The minimally damaged IJN CVs cold hae lanched a follow-p airstrike against the weakened USN. The Japanese wold thereby hae established control of the Coral Sea and been able to complete their inasion of Port Moresby. Discssion This paper deelops a seqential-fire ersion of the stochastic salo combat model by modifying the existing simltaneos-fire model. This modified model allows researchers to qantify the benefits of striking first, and yet it remains simple enogh to implement in serfriendly software sch as spread sheets. Nmerical tests indicate that the model prodces reslts ery similar to those of Monte Carlo simlation. The model is then sed to stdy the impact of different attack seqences in the 194 Battle of the Coral Sea. The reslts indicate that an American first strike wold hae been more beneficial for them than haing another aircraft carrier bt striking simltaneosly. Conersely, a Japanese first strike wold hae been deastating for the Americans and allowed the Japanese to inade Port Moresby. 11

12 Ftre mathematical research cold explore more sophisticated seqential models to see whether additional complexity wold proide any meaningfl improement in accracy. Ftre research cold also examine sitations where each side may be able to inflence (perhaps at some cost) the seqence of attacks; this wold inole adding elements of game theory, as in the attacker-defender game literatre. On the empirical side, researchers cold se the new model to stdy the naal battles at Midway in 194 and the Philippine Sea in 1944, where the airstrikes did occr seqentially. The model cold also be adapted for other contexts, sch as Connors (01) stdy of the 1853 Battle of Balaclaa or Edmiston s (011) stdy of a hypothetical near-ftre battle. Gien the ample empirical data that is aailable for the 1944 Ardennes campaign, there might be potential for stdy there as well (see, e.g., Hasken and Moxnes, 005). References Armstrong MJ (011). A erification stdy of the stochastic salo combat model. Annals of Operations Research 186 (1): Armstrong MJ (007). Effectie attacks in the salo combat model: salo sizes and qantities of targets. Naal Research Logistics 54 (1): Armstrong MJ (005). A stochastic salo model for naal srface combat. Operations Research 53 (5): Armstrong MJ (004). Effects of lethality on naal combat models. Naal Research Logistics 51 (1): Armstrong MJ and Powell MB (005). A salo combat analysis of the Battle of the Coral Sea. Military Operations Research 10 (4): Atkinson MP, Gtfraind A, Kress M (01). When do armed reolts scceed: lessons from Lanchester theory. The Jornal of the Operational Research Society 63 (10): Connors D (01). Into the Valley of Death Rode the Digital Six Hndred. Major research paper, Department of History, Brock Uniersity, St Catharines. Edmiston R (011). Palawan Scenario: Hghes Salo and Gaining Sea Speriority. Campaign analysis paper, Operations Research Department, US Naal Postgradate School, Monterey. Hasken K and Moxnes JF (005). Approximations and empirics for stochastic war eqations. Naal Research Logistics 5: Hasken K and Zhang J (01). The timing and deterrence of terrorist attacks de to exogenos dynamics. Jornal of the Operational Research Society 63 (6):

13 He F and Zhang J (01). Modeling 'Contracts' between Terrorist Grops and Goernments in a Seqential Game. Jornal of the Operational Research Society 63 (6): Hghes WP (000). Fleet Tactics and Coastal Combat, nd Ed. Naal Institte Press, Annapolis. Hghes WP (1995). A salo model of warships in missile combat sed to ealate their staying power. Naal Research Logistics 4 (): Johnson IR and MacKay NJ (011). Lanchester models and the Battle of Britain. Naal Research Logistics 58: 10-. Lcas TW and McGnnigle JE (003). When is model complexity too mch? Illstrating the benefits of simple models with Hghes' salo eqations. Naal Research Logistics 50: Lndstrom JB (1984). The First Team: Pacific Naal Air Combat from Pearl Harbor to Midway, Naal Institte Press, Annapolis. MacKay NJ (009). Lanchester models for mixed forces with semi-dynamical target allocation. Jornal of the Operational Research Society 60: Ross SM (1993). Probability Models, 5th ed. Academic Press, San Diego. Tiah YM (007). An Analysis of Small Nay Tactics Using A Modified Hghes Salo Model. Master s thesis, Operations Research Department, US Naal Postgradate School, Monterey. Zhang J and Bier VM (007). Balancing terrorism and natral disasters defensie strategy with endogenos attack effort. Operations Research 55 (5):

14 Table 1: Notation for deterministic model Red Ble Description A B Beginning force strength Offensie power per nit Loss sffered per hit w x Staying power per nit y z Defensie power per nit Table : Additional notation for stochastic model Red Ble Description n n Offensie missiles per nit per salo p p Probability of an accrate missile Off A Off B Total accrate offensie missiles per salo n y n z Defensie interceptions per nit per salo p y p z Probability of a sccessfl interception Def A Def B Total sccessfl defensie interceptions per salo Net BA Net AB Nominal non-intercepted offensie missiles F NetBA G NetAB cdf of nominal non-intercepted offensie missiles f NetBA g NetAB pdf of nominal non-intercepted offensie missiles A 1* B 1* Nominal sriing force strength after one salo F G cdf of nominal sriing force strength f g pdf of nominal sriing force strength A 1 B 1 Actal sriing force strength after one salo Table 3: Comparison of Ble losses, model erss simlation, for mean loss, standard deiation of loss (SD), probability of zero loss (P[0]), and probability of total loss (P[all]) Mean SD P[0] P[all] Regression R, model. simlation Regression slope, model. simlation Aerage of differences, model - simlation Variance of differences, model - simlation Greatest difference, model - simlation

15 Table 4: Coral Sea model inpts deried in Armstrong & Powell (005) USN IJN Nmber of aircraft carriers Bomber aircraft per carrier Fighter aircraft per carrier Probability of sccessfl deployment per bomber Probability of sccessfl intercept per fighter Aerage loss sffered per bomber Standard deiation of loss per bomber Table 5: Estimated Coral Sea losses from the model as the attack seqence and USN force size are aried. The nmbers show the mean loss, standard deiation of loss (SD), probability of zero loss (P[0]), and probability of total loss (P[all]), as well as the difference of means = (USN mean) (IJN mean). USN CVs Attack Seqence USN: Mean SD P[0] P[All] IJN: Mean SD P[0] P[All] Difference of Means 1 Simltaneos % 100% % 0% USN first % 9% % 0% USN second % 100% % 0% Simltaneos % 31% % 0% USN first % 0% % 0% +1.7 USN second % 31% % 0% Simltaneos % 0% % 84% USN first % 0% % 84% USN second % 0% % 0% Simltaneos % 0% % 100% USN first % 0% % 100% USN second % 0% % 50%

16 IJN mean loss - USN mean loss Probability from Model Figre 1: Scatter plot comparing model erss simlation for the probability that all Ble nits are eliminated. Each circle represents the reslts from one scenario, while the diagonal line indicates where the circles wold fall if the fit had been perfect Probability from Simlation Figre : Interaction plot for the difference between the IJN and USN mean losses from the model as the attack seqence and the USN force size are aried. Positie nmbers indicate a USN adantage USN first Simltaneos USN second Nmber of USN CVs in battle 16