Four people, Andrew Roberts reminds us in this survey of allied grand strategy in the Second World War, directed and won "the war in the West": President Franklin D. Roosevelt , Winston Churchill, the American Chief of Staff General George Marshall and his British opposite number, General Sir Alan Brooke.

The naval chiefs in both countries conducted their own wars, while the airmen loyally supported their military colleagues. Allied strategy was hammered out by continuous interaction between these four men, from their first cautious encounters in December 1941 until their acrimonious farewells in May 1945.

In 1941 the British and American leaders met virtually as strangers, and first impressions were not favourable. In the eyes of the Americans the British were snooty, cowardly and devious, concerned only with using American strength to pull their own imperial chestnuts out of the fire. The British saw their new allies as naive, ignorant and overbearing.

These stereotypes translated into strat­egic concepts. The British believed Marshall's desire to mobilise American strength to destroy the Wehrmacht in a decisive encounter in northern France to be foolishly naive. The Americans considered the British desire to delay this encounter until the Allies had gained control of the sea and air and diverted enough German strength to other fronts to make the odds sufficiently favourable to be timorous and "politically motivated". Round this simple theme the four leaders were to weave their variations for three and a half years while the Red Army was absorbing, and defeating, the German onslaught in the east.

The four made a fascinating quartet, and Roberts does full justice to their qualities and defects. Churchill's genius and eloquence made him the outstanding figure, and he can claim credit for the initial strategic concept: to give overriding priority to the war in Europe, to gain command of the sea and the air, to "close the ring" by obtaining control of the Mediterranean and only then attack , so it was hoped, a distraught and weakened foe.

Churchill was never enthusiastic about a landing in north-west Europe, and when the largely British victories in the Mediterranean opened up new prospects he embraced them with an enthusiasm that ignored all the agreements carefully hammered out by the Allied chiefs of staff. By the last year of the war the irresponsibility of his proposals was almost manic, as embarrassing to his military staffs as it was infuriating to his allies.

The man who had to bear the burden of Churchill's eccentricities was Brooke, and they nearly sent him mad. But for Roberts, Brooke is no hero. He does not buy the thesis that Brooke had from the beginning a coherent "Mediterranean strategy" that conformed with a traditional "British Way of Warfare". If anything drove Brooke's strategic concept it was the realisation that the British Army, having been driven from the Continent in 1940 and again, in Greece, in 1941, did not have a hope in hell of getting back in 1942, and even if they did they would again be defeated. They should therefore not even try until the odds were overwhelmingly in their favour, and meanwhile they should fight in the Mediterranean where the odds were favourable.

For Brooke, Marshall's fixation on a cross-­channel attack showed an inability to "think strategically". In this he was quite wrong. Marshall had at least as good a strategic mind as Brooke himself. But what Marshall did fail to do, at least initially, was to think operationally; that is, to analyse the successive military operations involved in implementing his strategy - operations whose complexity Brooke understood all too well from his own bitter experience.

If Roberts does have a hero it is Marshall. It was Marshall who insisted, over the opposition of the US Navy and much of American public opinion, that the war against Germany should be given priority over that in the Pacific, and that it could be won only in north-west Europe. While accepting reluctantly that concentration on that battlefield could be neither so immediate nor so total as he would have wished, he successfully resisted the increasing British pressure to convert operations initially acceptable as necessary diversions into rivals for priority. Marshall did well to accept Brooke's view in 1942 that a cross-­channel attack was for the time being out of the question; but he did equally well to overrule him a year or so later, when the demands of Mediterranean operations were making Brooke as well as Churchill weaken on their original ­commitment.

But Marshall could not have succeeded without the backing of Roosevelt; a man who knew nothing of strategy but a great deal about politics. Roosevelt backed him over the "Europe First" strategy but insisted that, if a cross-­channel operation was impossible in 1942, domestic pressure made it necessary to attack somewhere else; which left only the attack on French north Africa being urged by the British. It was then Roosevelt who backed him in restraining his allies from any further operations that would weaken the original concept, notably any exploitation of the Italian campaign after the fall of Rome. It may have been Churchill who proposed, but it was ultimately Roosevelt who disposed the strategy of the Allies.

Roberts is thorough and meticulous in his chronicle of this four-year dialogue: perhaps rather too thorough. His attempts to lighten his narrative with descriptions of the various conference venues and the personalities involved, together with his own ­perceptive comments and witty asides, cannot quite prevent these 600 pages from being rather heavy going.

But he is admirable for his honesty, rare in British historians, over two vital matters. The first is that in the Wehrmacht the Allies were facing an army very much better than their own - better equipped, better commanded, better trained, more highly motivated - and it continued to be so until the very last days of the war. Secondly, his Masters and Commanders may have "won the war in the West", as he puts it, but they did not win the war as such. Four out of every five of the Germans killed in combat died on the Eastern Front.

By diverting German military strength, and devastating German industry and cities with their bombing, the Western allies certainly made, as he puts it, "a vital contribution". But it was a contribution to a war being fought, and ultimately won, on the Eastern Front. It is time, after more than half a century, that we got this into proportion.