# Publications

The theory of tropical series, that we develop here, firstly appeared in the study of the growth of pluriharmonic functions. Motivated by waves in sandpile models we introduce a dynamic on the set of tropical series, and it is experimentally observed that this dynamic obeys a power law. So, this paper serves as a compilation of results we need for other articles and also introduces several objects interesting by themselves.

The paper applies Looking Forward Approach to analyze the world oil market with the framework of a differential game model of quantity competition oligopoly. Namely Looking Forward Approach is used to take into account dynami- cally updating information. Under the information we understand the forecast of the oil demand dynamics. We focus on the period from December 2015 to November 2016 and suppose that during this time interval countries did not cooperate offi- cially on the amounts of oil to be produced. Therefore, their behavior can be mod- eled using the non-cooperative game model. As a solution concept for this conflict- controlled process we use feedback Nash equilibrium. In order to define the pa- rameters of model open source data is used, results of numerical simulations and comparison with the historical data are presented.

We study stochastic voting models where the candidates are allowed to have any smooth, strictly increasing utility functions that translate vote shares into payoffs. We find that if a strict Nash equilibrium exists in a model with an infinite number of voters, then nearby equilibria should exist for similar large, but finite, electorates. If the votes are independent random events, then equilibria will not depend on the utility functions of the candidates. Our results have implications for existing models of redistributive politics and spatial competition, as the properties of pure-strategy equilibria in such games carry over to equilibria in games with arbitrary candidate preferences. On the other hand, candidate utility functions will matter if the individual voting decisions are correlated. In the presence of aggregate uncertainty, such as changing economic conditions or political scandals, the preferences of parties and candidates with respect to shares of votes will have an effect on political competition.

A mixed manna contains goods (that everyone likes), bads (that everyone dislikes), as well as items that are goods to some agents, but bads or satiated to others.

If all items are goods and utility functions are homothetic, concave (and monotone), the Competitive Equilibrium with Equal Incomes maximizes the Nash product of utilities: hence it is welfarist (determined utility-wise by the feasible set of profiles), single-valued and easy to compute.

We generalize the Gale-Eisenberg Theorem to a mixed manna. The Competitive division is still welfarist and related to the product of utilities or disutilities. If the zero utility profile (before any manna) is Pareto dominated, the competitive profile is unique and still maximizes the product of utilities. If the zero profile is unfeasible, the competitive profiles are the critical points of the product of disutilities on the efficiency frontier, and multiplicity is pervasive. In particular, the task of dividing a mixed manna is either good news for everyone, or bad news for everyone.

We refine our results in the practically important case of linear preferences, where the axiomatic comparison between the division of goods and that of bads is especially sharp. When we divide goods and the manna improves, everyone weakly benefits under the competitive rule; but no reasonable rule to divide bads can be similarly Resource Monotonic. Also, the much larger set of Non Envious and Efficient divisions of bads can be disconnected so that it will admit no continuous selection.

Non-transferable utility game of oil market is considered. Special approach for defining solution is used. This approach enables to construct a real time models of conflicting processes. Connection between the solution in the game with moving information horizon and solutions on the truncated time intervals is shown.

I consider the problem of allocating *N* indivisible objects among *N* agents according to their preferences when transfers are absent and an outside option may exist. I study the tradeoff between fairness and efficiency in the class of *strategy-proof* mechanisms. The main finding is that for *strategy-proof* mechanisms the following efficiency and fairness criteria are mutually incompatible: (1) *ex-post efficiency* and *envy-freeness*, (2) *ordinal efficiency* and *weak envy-freeness,* and (3) *ordinal efficiency* and *equal division lower bound*. Result 1 is the first impossibility result for this setting that uses *ex-post efficiency *; results 2 and 3 are more practical than similar results in the literature. In addition, for N=3, I give two characterizations of the celebrated random serial dictatorship mechanism: it is the unique *strategy-proof*, *ex-post efficient* mechanism that (4) provides agents that have the same ordinal preferences with assignments not dominated by each other (*weak envy-freeness among equals*), or (5) provides agents that have the same cardinal preferences with assignments of equal expected utility (*symmetry*). These results strengthen the characterization by Bogomolnaia and Moulin (2001); result 5 implies the impossibility result by Zhou (1990).

*A medium-scale nonlinear dynamic stochastic general equilibrium (DSGE) model was estimated (54 variables, 29 state variables, 7 observed variables). The model includes an observed variable for stock market returns. The root-mean square error (RMSE) of the in-sample and out-of-sample forecasts was calculated. The nonlinear DSGE model with measurement errors outperforms AR (1), VAR (1) and the linearised DSGE in terms of the quality of the out-of-sample forecasts. The nonlinear DSGE model without measurement errors is of a quality equal to that of the linearised DSGE model.*

Supposing that Player 1’s computational power is higher than that of Player 2, we give three examples of different kinds of public signal about the state of a two-person zero-sum game with symmetric incom- plete information on both sides (both players do not know the state of the game) where Player 1 due to his computational power learns the state of the game meanwhile it is impossible for Player 2. That is, the game with incomplete information on both sides becomes a game with incomplete information on the side of Player 2. Thus we demonstrate that information about the state of a game may appear not only due to a private signal but as a result of a public signal and asymmetric computational resources of players.

In this study, a novel approach for defining and computing a solution for a differential game is presented for a case, wherein players do not have complete information about the game structure for the full time interval. At any instant in time, players have certain information about the motion equations and payoff functions for a current subinterval, and a forecast about the game structure for the rest of the time interval. The forecast is described by stochastic differential equations. The information about the game structure updates at fixed instants of time and is completely unknown in advance. A new solution is defined as a recursive combination of sets of imputations in the combined truncated subgames that are analyzed by the Looking Forward Approach. An example with a resource extraction game is presented to demonstrate a comparison of payoff functions without a forecast and that with stochastic and deterministic forecasts.

New simple forms of deviation from rational expectations (RE) are suggested: temporary near-rational expectations (TNRE) and persistent near-rational expectations (PNRE). The medium-scale DSGE model was estimated with the RE, the TNRE and the PNRE. It was estimated with and without observations from the survey's expectations. The quality of the out-of-sample forecasts was estimated. It is shown that near-rational concepts produce the same advantages as learning, without its disadvantages (including the absence of ‘learning expectations’ reactions on policy change). The influence of the observed expectations on forecasting quality was analysed.