Tamtam Proceedings - lamsin

30 Habbaltheorem 1 There exists a Nash equilibrium, i.e. a pair of strategies (µ ⋆ , k ⋆ ) ∈ S 1 × S 2such thatµ ⋆ solves minµ∈S 1j 1 (µ, k ⋆ ) (5)k ⋆ solvesmin j 2 (µ ⋆ , k) (6)k∈S 2For numerical experiments, we considered the minimax (or duel) problemj 2 (µ; k) = −j 1 (µ; k) = −L 1 (ρ; p)which models a game where the first player wants to minimize the pressure drop, while onthe contrary the second player wants to maximize it (or, equivalently, wants to minimizethe drainage of the network).Such a game is also known as a zero-sum game.4. A Computational experimentThe domain Ω is a rectangle. The upper side is Γ V , and the lower is Γ T . Maximumallowed volume fractions are 40% for (TAF) and10% for (aAF).The Nash overall loop converged in three iterations ; each partial optimization took aroundthirty iterations (and two hundred FEM runs) to converge.Converging strategies are presented in figures 1-2-3.In figure-1, the first player (TAF), having the information that the second player (aAF)has played a uniform strategy, plays its optimal strategy which consists, as could be expected,in a single channel, preserving the volume constraint, located at the central lineof the rectangle (thanks to the symmetry of the problem). At its turn, the second player,informed that the first player has played a uniform strategy, simply puts as much antiangiogenicas possible around the tumor Γ T . Quite unexpectedly, (aAF) does not play auniform horizontal density, but creates a small excavation.Then, at the second Nash iteration, the first player knows that the second one has cutoff the way to the tumor, so it starts to develop alternative channels. At the same time,the second player knows now that the TAF has a strong presence within the excavation,which is then filled as shown in figure-2.The final iteration, yielding a numerical Nash equilibrium is shown in figure-3. The resultingporosity distribution in figure-3 does not exhibit any arterial-tree branching structure,but only multiple channels.Multiple channels seem to be the best response of the activators to optimally distributedinhibitors.TAMTAM –Tunis– 2005