Hawks, doves, Israel and Iran

In The Conversation last October, Andrew Thomas (Deakin University) discussed the recent (at that time) military flare-up between Iran and Israel, likening it to a 'game of chicken':

Israel’s strike on military targets in Iran over the weekend is becoming a more routine occurrence in the decades-long rivalry between the two states...

There is a reason why direct military strikes between nations are rare, even between sworn enemies. When attacking another state, it is difficult to know exactly how they will respond, though a retaliatory strike is almost often expected.

This is because defence forces are not just used for fighting and winning wars – they are also vital to deterring them. When a fighting force is attacked, it’s important for it to strike back to maintain the perception it can deter future attacks and make a display of its capabilities. This is what is happening right now between Israel and Iran – neither side wants to appear weak.

If this is the case, where does the escalation end? De-escalation is essentially a game of chicken – one side has to be content with not responding to an attack to take the temperature down.

My ECONS101 class has been covering game theory this week, including the chicken game. In the traditional game of chicken there are two rivals in cars, one at each end of the same street. They drive towards each other at top speed, and whichever rival swerves away first loses the game. So, each rival can choose to speed ahead or swerve away, and each would prefer to speed ahead and win the game. However, the problem is that if both simply keep speeding ahead, it will end in a disastrous crash.

I also recently read the book Hidden Games, by Moshe Hoffman and Erez Yoeli (which I reviewed here). Hoffman and Yoeli have an interesting section in the book on the hawk-dove game, which essentially the chicken game but with a slightly different motivating context. In the hawk-dove game, two rivals are competing over some resource. Each rival can choose to be aggressive or submissive, and whichever rival is more aggressive will win the resource. Each rival would prefer to be aggressive and win the resource. However, if both are aggressive, it ends in a massively disastrous battle.

Coming back to the case of Iran and Israel, this is clearly an example of the hawk-dove game (or the chicken game, if you prefer). This game is laid out in the payoff table below, where the strategies for Israel and Iran are to be aggressive, or submissive. The payoffs are expressed as "+" for good outcomes, and "-" for bad outcomes (and "--" is particularly bad), while zero is a neutral payoff.

To find the Nash equilibrium in this game, we use the 'best response method'. To do this, we track: for each player, for each strategy, what is the best response of the other player. Where both players are selecting a best response, they are doing the best they can, given the choice of the other player (this is the definition of Nash equilibrium). In this game, the best responses are:

1. If Iran chooses to be aggressive, Israel's best response is to be submissive (since "-" is better than "--" as a payoff - in other words, taking a bit of punishment is better than a massively disastrous war) [we track the best responses with ticks, and not-best-responses with crosses; Note: I'm also tracking which payoffs I am comparing with numbers corresponding to the numbers in this list];
2. If Iran chooses to be submissive, Israel's best response is to be aggressive (since "+" is better than "0" as a payoff);
3. If Israel chooses to be aggressive, Iran's best response is to be submissive (since "-" is better than "--" as a payoff); and
4. If Israel chooses to be submissive, Iran's best response is to be aggressive (since "+" is better than "0" as a payoff).

In this scenario, there are no dominant strategies. Neither country has a strategy that is always better for them, no matter what the other country chooses to do. However, there are two Nash equilibriums (outcomes where both players are playing their best response), which occur when one country is aggressive, and the other is submissive.

The thing about the Iran-Israel hawk-dove game is that it isn't really a simultaneous game, as shown in the table above. It is a sequential game. Each player chooses whether to be aggressive or submissive, knowing what the other player chose to do previously. That sequential game is shown below. [*]

We can solve a sequential game using 'backward induction', which is essentially the same as the best response method, except we make sure we start with the last player, and work our way backwards through the game to work out what the first player should do. The resulting equilibrium that we find will be a 'subgame perfect Nash equilibrium'. In this case:

1. If Israel chooses to be aggressive, Iran's best response is to be submissive (since "-" is better than "--" as a payoff) - now, since Iran would never choose to be aggressive when Israel has already been aggressive, Israel knows that the outcome will be that Iran is submissive;
2. If Israel chooses to be submissive, Iran's best response is to be aggressive (since "+" is better than "0" as a payoff) - now, since Iran would never choose to be submissive when Israel has already been submissive, Israel knows that the outcome will be that Iran is aggressive;
3. Israel will choose to be aggressive (since "+" is better than "-" as a payoff).

The subgame perfect Nash equilibrium is that Israel is aggressive, and Iran is submissive. As it turns out, that's sort of what happened. After an initial flurry of missile attacks, Iran stopped escalating the conflict.

One last thing to note is that this is a repeated game. Israel and Iran will find themselves in conflict often. The games as outlined above suggest that whichever country is the initial aggressor will end up getting their way, because the country moving second in the sequential game will be better off being submissive than retaliating. However, in repeated games, the outcome can often deviate from the equilibrium for strategic reasons.

Israel clearly doesn't want Iran to be aggressive, even though aggression would be good for Iran, if they moved first in this game. So, Israel wants to convince Iran not to make an aggressive first move. The only way that Israel can do that is to convince Iran that Iran would be worse off by making an aggressive first move. Israel needs to convince Iran that Israel will always retaliate with aggression. That would deter Iran from being aggressive as a first move. How does Israel achieve this? By developing a reputation for aggressively retaliating against any aggression. And indeed, that is what Israel has done (one need only look at Gaza or Lebanon for confirmation of this).

Israel's aggressive response to attacks by Iran, Gaza, and Lebanon is part of a strategic plan to deter future aggression against Israel. Many of us may not like it, but it's strategically rational. Whether it has a lasting effect remains to be seen.

*****

[*] I'm showing the game as having Israel move first. However, if you read Thomas's article, you'll see that 'who started it' is actually contested. I'm not taking a stand on that here, and in fact the game looks identical if Iran moves first.