Edit: This post is meant for fun: I was initially interested in seeing a wider range of mathematical techniques applied to computing the likelihood of a Deal or No-Deal outcome of the negotiations. Specifically, I was interested in how mathematics can be applied to a current real-world example, wanting to see how different fields of mathematics might approach the problem (i.e. Bayesian probabilists vs. Game Theorists).
Since there have been no answers so far, I have decided to narrow the "question" specifically to Game Theory, as I would be most interested in seeing how this particular discipline could be applied here. If any game theorists are interested in responding, I would be very interested in seeing an example.
Original post: This post is meant for fun: given that we have two days of negotiations left, use any mathematical technique or reasoning to compute the probability of a No-Deal. Be creative: game theory, probabilistic theory, Monte Carlo: it's all allowed.
I will post my own answer in a few hours.
As per the request in the comments below, here are some details:
- Negotiations must end by Sunday, whether a deal has been reached or not (this post is written on Fri evening, i.e. two days before the deadline)
- EU wants guarantees on aligned regulation ("level playing field"), to make sure that UK can't change it's regulatory environment in key areas to "undercut the EU" and gain a competitive advantage
- UK rejects the above, because it argues the point of Brexit is to be able to pass its own laws (i.e doesn't wanna be kept in the EU's "regulatory orbit")
- Both sides have much to loose: EU exports many goods to the UK (trade between Germany and UK alone was worth $139bn in 2019), whilst the goods exported by the UK to the EU amounted to £294bn in 2019 (43% of all UK exports): no deal means trade tariffs governed by WTO (mathematically, treat no deal as economically damaging to both sides)
- A deal would be economically beneficial to both sides, but EU demands regulatory alignment, which is rejected on political grounds by the UK (and there is an argument for economic benefits in the longer run if own laws can be passed in the future)
Mathematically, you can look at current odds (and their evolution over the past week and months), current market moves in the GBP currency, or use other techniques: use anything you like to come up with a mathematically-backed answer.
Try to stay apolitical.
Best Answer
Initial super-basic model to get the thread started (I hope to see some Game theorist & Bayesian probabilist contributions!)
Event $A$: EU willing to offer trade deal without UK fully committing to regulatory alignment
Event $B$: UK willing to accept a trade deal whilst agreeing to commit to regulatory alignment
Event $C$: No-Deal Brexit
Event $C^{'}$: Some sort of Deal
Now:
$$\mathbb{P}(C')=\mathbb{P}(A)+\mathbb{P}(A'\cap B)=\\=\mathbb{P}(A)+\mathbb{P}(A')\mathbb{P}(B)=\\=\mathbb{P}(A)+(1-\mathbb{P}(A))\mathbb{P}(B)$$
My subjective guesstimate is that $\mathbb{P}(A)=0.15$, whilst $\mathbb{P}(B)=0.35$. Then clearly:
$$\mathbb{P}(C')=0.15+0.85*0.35=0.4475$$
So that would mean:
$$\mathbb{P}(C)=1-0.4475=55.25\%$$
Latest odds of "no deal" are now $8/11$, which implies: $\mathbb{P}(C)=\frac{8}{8+11}=42.1\%$.
Clearly my very basic model is off. I might want to do some Bayesian refinements and assume distributions of probabilities, instead of constants, and regress these on latest events (as proposed in the comments in the original question: might have a shot 2mrw!).
I am sure a game theory answer can beat my initial super-basic model!