Here is the Saudi-Israel game framework as a Stack Exchange question with MathJax:

game-theory markov-processes

How can I model the dynamics of Saudi-Israel relations using a stochastic game with Markov states?

I'm trying to analyze the fluctuations in Saudi-Israel relations over time using a game-theoretic model. Here is a basic framework I've devised:

\begin{align} \text{Players}: && S = \text{Saudi Arabia}, I = \text{Israel} \\ \text{States}: && H = \text{Hostile}, C = \text{Cooperative} \\ \text{Payoffs per period}: && H &= (U(SD), U(ID)) \\ C &= (U(SC), U(IC)) \\ \text{State transitions}: && H \to C &= P(\text{Bincrease}) \text{ for } S, P(\text{Bdecrease}) \text{ for } I \\ C \to H &= P((\text{Z,Rincrease})) \text{ for } S, P((\text{Z,Rdecrease})) \text{ for } I \end{align}

How can I implement and analyze this type of model? Some questions I have:

  • How should I define the payoff functions $U(.\cdot)$ and transition probabilities $P(.\cdot)$ based on factors like cooperation benefits/costs and conflict risks?
  • How can I simulate the model to explore how different parameters/assumptions impact the time spent in hostile vs. cooperative states?
  • What kinds of insights might be gained from such a model? What are its limitations in analyzing real-world relationships like this?

How do I model such stochastic dynamic games?

  • 1
    $\begingroup$ Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. $\endgroup$
    – Community Bot
    Commented Mar 11, 2023 at 16:33

1 Answer 1


Here's an attempt of an analytic, SDE, to frame an insight.

$dX_t = [A\_1(X_t)*(U(SC) - U(SD)) + A\_2(t)*[P(Z,R\_{increase}) - P(Z,R\_{decrease})]] dt + [B\_1*dv(oil\_price) + B\_2*dv(petrodollar\_status)] dW_t$


  • X$_t$ is the state variable representing relations
  • $A\_1$ and $A\_2$ are functions for drift towards cooperation vs. conflict, dependent on current state and time-varying conditions
  • $B\_1$ and $B\_2$ are parameters determining diffusion magnitude from oil price and petrodollar volatility
  • $dv(.)$ represents volatility of the economic variables
  • $dW_t$ is a Wiener process representing random shocks

The specific forms of the $A_1$, $A_2$ and volatility functions, and values of the parameters, would be estimated based on data/modeling assumptions about the relationships between the variables.

Simulating and analyzing the SDE could reveal how relations might fluctuate around trends, and be impacted by economic factors.

Of course, this is a simplified model and would face the usual challenges in capturing such a complex real-world system.


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