We consider multistage bidding models where two types of risky assets (shares) are tradedbetween two agents that have different information on the liquidation prices of tradedassets. These prices are random integer variables that are determined by the initialchance move according to a probability distribution p over thetwo-dimensional integer lattice that is known to both players. Player 1 is informed on theprices of both types of shares, but Player 2 is not. The bids may take any integer values.The model of n-stage bidding is reduced to a zero-sum repeated game withlack of information on one side. We show that, if liquidation prices of shares have finitevariances, then the sequence of values of n-step games is bounded. This makes itreasonable to consider the bidding of unlimited duration that is reduced to the infinitegame G∞(p). We give the solutions for thesegames. Optimal strategies of Player 1 generate random walks of transaction prices. Butunlike the case of one-type assets, the symmetry of these random walks is broken at thefinal stages of the game.

A matrix game is played repeatedly, with the actions taken at each stage determining both a reward paid to Player I and the probability of continuing to the next stage. An infinite history of play determines a sequence (Rn) of such rewards, to which we assign the payoff lim supn (R1 + · ·· + Rn). Using the concept of playable strategies, we slightly generalize the definition of the value of a game. Then we find sufficient conditions for the existence of a value and for the existence of stationary optimal strategies (Theorems 8 and 9). An example shows that the game need not have a value (Example 4).