Washington ? Perhaps it was the “fog of simulation.” But the scariest aspect of a U.S.-Iran war game staged this week was the way each side miscalculated the other’s responses — and moved toward war even as the players thought they were choosing restrained options.

The Iran exercise was organized by Kenneth Pollack, a senior fellow at the Brookings Institution’s Saban Center for Middle East Policy. It included former top U.S. officials as Washington policymakers, and prominent Iranian-American experts playing Tehran’s hand. I was allowed to observe, on condition that I wouldn’t name the participants.

The bottom line: The game showed how easy it was for each side to misread the other’s signals. And these players were separated by a mere corridor in a Washington think tank, rather than half a world away.

Misjudgment was the essence of this game: Each side thought it was choosing limited options, but their moves were interpreted as crossing red lines. Attacks proved more deadly than expected; signals were not understood; attempts to open channels of communication were ignored; the desire to look tough compelled actions that produced results neither side wanted.

Let’s walk through the simulation to see how the teams stumbled up the ladder of escalation. The game was set in July 2013, with some broad assumptions: It was assumed that President Obama had been re-elected, the P5+1 negotiations remained deadlocked and Israel hadn’t launched a unilateral attack.

The game controllers added some spicy details: assassinations of Iranian scientists were continuing; and the U.S., Israel and Britain were developing a new cyber weapon (imaginary code name: NATIONAL PASTIME) to disrupt power to Iran’s nuclear and military facilities. Even so, the Iranian supreme leader thought America was a paper tiger, telling aides: “The Americans are tired of the fight and they are led by a weak man with no stomach for the struggle.”

Meanwhile, Iran was pushing ahead with its nuclear program; it had a rough design for a weapon and in three to four months would have enough highly enriched uranium to make two bombs.

The action started on July 6 with an Iranian terror operation: A bomb destroyed a tourist hotel in Aruba, killing 137 people, many of them Americans, including a vacationing U.S. nuclear scientist. The damage at the hotel was far greater than the Iranians had expected.

The U.S. team recommended strong retaliatory moves to signal Iran that it had crossed an “unacceptable threshold.” The U.S. bombed a Revolutionary Guards camp in eastern Iran; launched a cyber attack that disrupted power at 40 Iranian security facilities; and warned Iranian operatives in 38 countries that they were known and vulnerable. U.S. military leaders in the game complained that these calibrated moves were halfway measures.

Bombing the Iranians’ homeland rocked their team. It crossed a red line, in a way the U.S. side hadn’t anticipated. The Tehran players spurned a secret message from Obama, delivered through Russia, warning of “dire consequences” if the nuclear program wasn’t stopped; the imaginary Iranian defense minister called it a “bluff.” The Iranians wanted to respond forcefully, but not so much they would trigger an attack on their nuclear facilities.

Then the Iranian team made what proved a devastating mistake. After rejecting the most aggressive options (such as attacking Fifth Fleet headquarters in Bahrain), they chose limited actions that were described as “random mining” of the Strait of Hormuz and “harassment” of U.S. ships in the Gulf. The Iranians also dispersed their stockpile of uranium, but only half, to signal they were still willing to negotiate. But the U.S. missed the message.

“They’ve crossed our red line,” responded the imaginary U.S. national security adviser — expressing the group’s mistaken view that the Iranians had decided to close the strait and attack U.S. vessels. As tensions increased, oil prices headed toward $200 a barrel.

U.S. military options were between harsh and harsher: (a) reopen the Strait by force and deliver an ultimatum that Iran stop its nuclear program within 24 hours; or (b) hit Iran’s nuclear facilities simultaneously with reopening the Strait. Military logic seemed to require the strongest move. The U.S. team ultimately voted, five to three, for an attack across Iran to disable the nuclear program and destroy coastal defenses.

The unsolved puzzle for the U.S. side was how to stop the conflict, once it started. The Iranians, for their part, had decided to bleed the U.S. in a protracted struggle. The lesson of the exercise, concluded Pollack, is that “small miscalculations are magnified very quickly.”