As the great accounting fight of the past decade moves near to a conclusion that seems likely to satisfy no one, the cast of characters is suddenly changing. The two most important accounting bodies in the world will be under new leadership as they try to move on from bruising battles that have pitted them against banks, bank regulators and politicians.

The changes are coming before it is clear just how far the two boards — the Financial Accounting Standards Board, which sets accounting rules in the United States, and the International Accounting Standards Board, whose rules are followed in a growing number of countries — will have to back down in their efforts to force banks to report the current value of their assets.

Both boards have made concessions that alarmed some believers in the use of market values, but neither has fully satisfied powerful critics in the banks, who have gathered support from bank regulators and politicians on both sides of the Atlantic.

Robert H. Herz, the chairman of the American group since 2002, stepped down at the beginning of this month. His decision, announced in late August, came as a surprise to many accountants. A replacement for David Tweedie, the chairman of the international group since 2001, was picked last week, although Mr. Tweedie’s term will not end until mid-2011.

The new leaders will play a major role in shaping accounting rules around the world. It will become clear how successful the two boards are being at producing “convergence” of their rules, and decisions will be made by the Securities and Exchange Commission on whether to allow international rules to be used by American companies instead of the traditional generally accepted accounting principles, or GAAP, which are set by the American board.

Differing views of the proper role of accounting rule makers around the world are at stake. In the United States, there is general support for the concept that rules should be set by independent experts, although that has not kept politicians from intervening at times, as they have in the fair-value debate. That idea of using market values is generally supported in Britain but has much less support in continental Europe, particularly in France.

Some American supporters of independent rule-making fear that a decision to move to the international rules may mean making unwelcome compromises. Some in Europe fear — and others hope — that American involvement will restrain European politicians from exercising the influence they desire.

The arguments over the use of market value — or fair value, as the accounting rules characterize the numbers — have polarized accounting debates. Banks complained bitterly that the limited rules forcing the use of market value for some assets contributed to the financial crisis, first by exaggerating their wealth as prices rose and then by making them appear worse than they really were when market prices plunged to unreasonably low levels.

That argument was exaggerated. The securities on which the banks lost the most money — largely mortgage-backed bonds — had not soared in price, and so had not made the banks look better. But banks had been so confident that the securities could not lose value that they borrowed heavily to own more and more of them, leaving them vulnerable to the collapse. When that came, however, there were no buyers. Many of the few trades that were made did take place at distressed prices.

The first assault by banks on fair value was aimed at preventing them from having to recognize such distressed prices as being fair, and they largely succeeded in that as both boards hurriedly put out advice under political pressure. The American board acted after Mr. Herz was called to a Congressional hearing and was berated by legislators from both parties.

After that, under heavy pressure to act quickly, the two boards chose differing courses. The international board adopted a new rule in record time concerning which assets had to be carried at market value. It generally allowed banks to ignore market values of loans, as they wished. But the board put off some other issues.

The American board put forth a stronger and more comprehensive proposal. That proposed rule would allow banks to keep losses from falling loan values off their income statements, but the banks would be forced to show market-value numbers prominently. Banks, and their regulators, complained loudly.

Mr. Herz told me this week that his decision to step down was not forced, but it appeared to have cleared the way for the board to soften its proposal. His temporary successor, Leslie F. Seidman, had opposed issuing the proposed rule. The board recently said that its consultations with investors and analysts — known as users of financial statements — had shown splits on the issue, with bank analysts more likely to go along with the companies they covered.

Some who question fair value worry that determining the figures will be difficult and ultimately subjective, at least for assets that are not readily traded. At the same time, there is little doubt that the actual value of an asset, not what was paid for it, is usually more relevant in evaluating the financial health of a company. “It is better to be vaguely right than to be precisely wrong,” Roman Weil, an accounting professor at New York University, said at an N.Y.U. conference this week.

Banks and bank regulators have frequently complained that market-value accounting is “pro-cyclical,” meaning it makes banks seem richer, and therefore more willing to lend, during booms, when restraint might be appropriate, and less willing to lend during busts, when the economy may need more lending.

Mr. Herz has repeatedly argued that such decisions are properly made by bank regulators, but that accounting should reveal the actual facts, not sugar-coated ones. At the same N.Y.U. conference, he said the account statements he received from Merrill Lynch during the financial crisis were also pro-cyclical, since they made him and his wife feel poorer and less willing to spend. Perhaps, he said, Merrill should have been told to send out more palatable numbers.

It now seems likely that the American board will back off at least part of the way from its proposal, although it is not clear how far it will go. The foundation that appoints members of the American board has decided to expand it to seven members, raising the possibility that new members might be more amenable to compromises. Ms. Seidman wants to stay as head of the board, but no decision on that will be made until early next year.

Arthur Levitt, a former chairman of the S.E.C., said in an interview this week that he feared “independent standard setting is in greater jeopardy than at any time since the F.A.S.B. was formed.” He added, “I don’t see any power at all on the side of fair value. Because of the power of the banks and the actions of the Congress, the battle is well on its way to being decided.”

Mr. Levitt knows about political pressure. In the 1990s, when he was chairman of the S.E.C., Senator Joseph Lieberman of Connecticut threatened to legislate against F.A.S.B. if it did not back down on efforts to force companies to show the value of stock options granted to employees as an expense. Mr. Levitt quietly advised the board to retreat, which it did, and that rule was not adopted for another decade. He later said that was the worst decision he had made as chairman.

The new chairman of the international board will be Hans Hoogervorst, a Dutch regulator and a former government minister. He has also been chairman of a group that monitored the international board on behalf of governments, but he is not an accountant and not an expert on technical rules. An Australian accountant, Ian Mackintosh, who is currently chairman of the British accounting standards board, was chosen as vice chairman.

The international board under Mr. Tweedie was at times very controversial, particularly in France, where bankers were angry over earlier efforts to increase the use of fair value.

In 2003, Claude Bébéar, then the chairman of AXA, the French insurer and perhaps the most influential business leader in the country, called Mr. Tweedie “a super-super-ayatollah” who “has a fascination with market value.”

European politicians expressed horror that the board was not answerable to democratic bodies. The European Commission asserted the right to approve any new accounting rule before it took effect in Europe, and it ruled that companies could ignore part of a rule accounting for derivative securities, something French banks have done. The commission also appointed a group, known as the European Financial Reporting Advisory Group, to advise on whether new rules were warranted.

The chairwoman of that group, Françoise Flores, a French accountant, said after Mr. Hoogervorst was named that “this appointment will help strike a good balance between the accountability and the independence of the international standard setter.”

Such comments bothered some Americans. “It is a strange balance,” said Stephen Zeff, an accounting professor at Rice University who has written extensively on international accounting rule-making. The board, he said, should be accountable to the users of financial statements, and for that independence was crucial.

In an interview, Mr. Hoogervorst tried to stake out a middle ground. He said it was “very clear that fair-value accounting has done a lot of good things. It has brought a lot of transparency. It forces financial institutions to recognize problems as soon as possible.” Had Japanese banks done that after the country’s asset bubble burst two decades ago, he said, they would have been better off.

“At the same time,” he added, he approved of “the pragmatic approach the I.A.S.B. ultimately took.”

Mr. Hoogervorst served as co-chairman of a body that advised the two accounting boards on the financial crisis, and concluded that a large part of the problem was that the banks had been allowed by regulators to take on too much leverage during the good time.

“Essentially, the banking industry is a welfare state within the market economy,” he said in a speech this year. “The main difference with the normal welfare state is that the benefits are very high and that they are usually determined by the recipients themselves.”