Repeated bidding games with incomplete information and bounded values: on the exponential speed of convergence
We consider the repeated zero-sum bidding game with incomplete information on one side with non-normalized total payoff. De Meyer, Marino (2005) and Domansky, Kreps (2005) investigated a game $G_n$ modeling multistage bidding with asymmetrically informed agents and proved that for this game $V_n$ converges to a finite limit $V_\infty$, i.e., the error term is $O(1)$. In this paper we show that for this example $V_n$ converges to the limit exponentially fast. For this purpose we apply the optimal strategy $\sigma_\infty$ of insider in the infinite-stage game obtained by Domansky (2007) to the $n$-stage game and deduce that it is $\varepsilon_n$-optimal with $\varepsilon_n$ exponentially small.
The paper considers a game-theoretical model of bidding with asymmetric information. One player has the inside information on the liquidation price of risky asset. The model is formalized with the repeated game with incomplete information on the side of uninformed player. We consider the case of external stopping of the game at the random moment. Insider's expected profit in the game of random duration if she applies the strategy optimal in infinite-stage game is obtained. This result allows to calculate the loss of insider in case of sudden disclosure of his private information.
We consider a discrete model of insider trading in terms of repeated games with incomplete information. The solution of the bidding game of beforehand unlimited duration was obtained by V. Domansky (2007). Insider's optimal strategy in the infinite stage game generates the simple random walk of posterior probabilities over the lattice l/m, l=0,...,m with absorption at the extreme points 0 and 1 and provides the expected gain 1/2 per step to insider. In this paper we calculate insider's profit in the game of any finite duration when he applies the strategy above. It is shown that this strategy is his epsilon-optimal strategy in n-stage game, where epsilon decreases exponentially. This means that the sequence of n-stage game values converges to the value of infinite game at least exponentially. The result obtained is interpreted as the loss of insider in the case of sudden disclosure of his private information. For the special case we compare obtained insider's profit with the exact game value (result of V. Kreps, 2009) and demonstrate that error term in the case of optimal insider's behaviour also decreases exponentially.
We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.
We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.
We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.