Mikhail Gorbachev has given President Reagan the answer that George Bernard Shaw gave to Mrs. Patrick Campbell when she proposed publishing their correspondence: "Why should I play horse to your Lady Godiva?"

A U.S. summit would have been a bonanza for a beleaguered president. But Gorbachev broke his promise to come here, perhaps after watching Reagan's news conference and realizing how desperately the president needs a turn as host and tour guide of the world's leading Marxist.

Towing Gorbachev around the wonders of the capitalist system -- perhaps a pig farm, a supermarket, Disneyland -- would do nothing for the Dow, nor would it dilute the bitterness of reverses on Bork and the contras. But it would have been a superb photo opportunity, and could have given the country a little hope and circus at a low moment.

But watching Reagan, Gorbachev may have seen that the president could, in the months ahead, become even hungrier for a summit. Gorbachev craves an arms treaty, even one like the "virtually" completed INF accord which is favorable to the United States.

Gorbachev's economic situation impels him to do almost anything to reduce crushing military budgets. But he learned from the news conference that there is nothing in the summit for him.

He also learned that while the leader of the capitalist world has grave economic problems, too, he is unwilling to admit it. Reagan will make the $23 billion deficit cuts forced by Gramm-Rudman-Hollings, but on raising new revenues, he seems to think that all will be well if he charges motorboat operators for the use of certain waterways. Wall Street emphatically disagrees.

Nor will Reagan admit how valuable a summit is to him. He will not trade it for "Star Wars."

Gorbachev wants at least to discuss the limits of Strategic Defense Initiative (SDI) testing. During his September visit, Soviet Foreign Minister Eduard Shevardnadze made two modest proposals about negotiating technical and scientific limits on testing the components. Or, failing that, how about observing the ABM treaty, as originally understood and ratified, for the next 10 years?

Secretary of State George P. Shultz took Reagan's nyet on both counts to Moscow with him.

Reagan at his news conference spoke of Star Wars in terms that said unmistakably that he is as obdurate about the system as he was four years ago when he first announced it. Neither scientific skepticism nor suffocating deficits have altered his views by a whit. Cutting back on SDI is as unacceptable as raising taxes.

Gorbachev knows that many Americans do not share what Sen. Sam Nunn (D-Ga.), the stalwart on defense, calls the president's "dream-world definition" of what SDI can accomplish. Nunn recently helped pass a measure that requires the president to consult the Senate if he puts into effect the "reinterpretation" of the ABM treaty that permits virtually unlimited testing of SDI systems. The president is undeterred, however.

The administration is now engaged in strenuous efforts to reinterpret Gorbachev's cancellation. They say it hurt him more than Reagan. But Shultz's bleak Reykjavik face told the story. It is a crushing disappointment. Another hope for serious negotiations, even strategic reductions, has broken on the rock of Reagan's Star Wars' fixation.

Gorbachev will pay for breaking his promise. Americans, deprived of yet another federal television spectacular, will be sore at him. The allies, despite their antipathy towards the INF treaty, feel let down. The White House is bravely saying he will come crawling back, that there will be a summit before the end of the year.

Only the right wing is happy. Gorbachev has given them the chance to renew their cry that you can't trust a Russian. It probably won't rid them of the INF treaty, which they hate with a great passion as they do any accord with the Soviets, but they will be spared the hoopla of the world's leading Marxist being made much of by people the right think should be plotting his downfall.

Reagan's news conference may have further convinced Gorbachev that Reagan might not be fun to talk to. The president was not at his best. He showed a desire to relive the past, to blame others for the mess at his door. He displayed no magnanimity to members of Congress he had grudgingly agreed to meet. He refused to show any cards in the imminent exchange with his "equal partners." A stubborn old party, Gorbachev probably thought.

If he treats his countrymen that way, Gorbachev may have reasoned, what kind of a time could I, a representative of the Evil Empire, expect?