Using game theory to analyse the current US-Ukraine-Russia interaction

 Yesterday, we witnessed unprecedented scenes in the White House's Oval Office.

The fireside chat of Presidents Trump and Zelensky dissolved into a spat that led to the cancelling of the signing of a deal between the US and Ukraine (the action starts at around minute 40 in the below video).

Below is an attempt to apply the concepts of Nash equilibrium, empty threats, credibility and commitment from game theory to shed some light on this complex situation.

We can probably simplify the current interaction with the below decision tree:

The payoffs are shown at each terminal node, with the order (US payoff, Russia payoff, Ukraine & EU payoff).

The idea here is that if there is no agreement between the US and Russia, the war will continue and all parties will lose somewhat.

If a ceasefire deal is reached, there is a fear that Russia may restart its aggression in 4-5 years, having recovered some of its military strength, which would be a disaster for Ukraine and the EU.

Can you work out the Nash equilibrium with the above structure?

Starting from the last move, according to this interpretation of the incentives, Putin's administration would prefer to renege on the ceasefire deal in 4-5 years and restart its aggression (getting 10 utils instead of 0).

Going back one node, which refers to Russia's choices today, Russia would prefer to pursue a deal, rather than continue with the aggression, in order to recover its military and economic strength (and get eventually a payoff of 10, rather than -5).

In the first node, Trump's administration would prefer to pursue a deal with Russia, even if it eventually reneges, as in the meantime it can enjoy the credit of having stopped a bloody conflict and prevented a nuclear disaster (payoff of 0, rather than -5).

The Nash equilibrium in this game is that the US pursues a deal, Russia agrees, but reneges in 4-5 years.

This is the worst outcome for Ukraine and the EU (-10 utils, as opposed to -5 if the war continues or 0 if there is a permanent ceasefire along the current line of contact), based on the assumption that Russia's military will progress stronger after a pause of 4-5 years.

Now, what can Ukraine and the EU do to avoid this scenario?

One demand that they have is to be part of the negotiations process. How does this change the outcome?

Consider the below decision tree.

As in the previous case, at the last node Russia would prefer to renege on the ceasefire deal after 4-5 years.

However, now Ukraine and the EU would choose to kill the deal and receive a payoff of -5 rather than -10. So the Nash equilibrium in this case is that no ceasefire deal is reached.

The Trump administration would like to avoid this outcome, so it tries to keep Ukraine and the EU out of the negotiations with Russia. President Trump insists that him being in office is a sufficient guarantee for Russia complying with the deal, but Ukraine and the EU see this as an empty promise, rather than as a credible threat to Russia.

The best outcome (from the above options) for Ukraine and the EU is to achieve a permanent ceasefire deal and receive a payoff of 0 rather than -5. How can this be achieved?

Here is where the demand for solid security guarantees comes in play. President Zelensky insists on receiving such guarantees, before agreeing on any deal.

These guarantees would greatly increase the cost for Russia to renege on the ceasefire agreement in 5 years (payoff at -10, rather than 0 with the ceasefire agreement), but they also come at a cost for the Trump administration, changing the payoffs as in the below tree:
  

 

With this tree, the Nash equilibrium is for Russia to comply with the ceasefire agreement (0 utils rather than -10) and in the previous node for Ukraine and the EU to agree with the deal (receive 0 utils rather than -5). 

With these payoffs, the US appears to be indifferent between pursuing a deal with the provision of strong security guarantees to Ukraine and having no ceasefire deal (-5 utils in each case).

The side deal that gives preferential rights to the US for the exploitation of Ukraine's mineral wealth may help with improving the payoffs for the current US administration, resulting in the below tree, where a permanent ceasefire is the game's only Nash equilibrium in pure strategies.

Here, US gets 0 utils with the permanent ceasefire deal, as the positive publicity from the Ukraine minerals deal compensates for the publicity cost of committing to Ukraine's long-term security, making it preferable to the no ceasefire outcome (0 rather -5 utils). Ukraine and the EU lose somewhat (-2 utils), but they are still better off compared to the current situation (-5 utils).

The above quick and rough analysis shows how game theory concepts could help us analyse complex negotiations and even find a path to peace.