Through history awards for crime were frequent. Experience teach us that through history in many countries results of sanction impositions did not bear fruit. Through two people game in game theory we showed theoretical explanation of mentioned terms and overcame disadvantages of sociological and economic access to crime. Through the game of criminals and police which we defined with payoff matrix Table 2.1 we assumed that criminals love to break the law if police do not enforce it and that criminals will abide the law if police enforce it. The other two assumptions refer to the fact that the police will not enforce the law if the criminals do not break it but they will prefer law and arrest enforcement if the criminals break it. Under these assumptions we cannot calculate pure Nash equilibrium and therefore the only game equilibrium under this assumptions is the mixed Nash equilibrium. We showed that expected penalty enlargement for criminals does not affect the law breaking intensity. Also we showed that expected penalty enlargement reduces law enforcement intensity. We approached the mixed Nash equilibrium finding problem through five different models where in first model we had rational approach with perfect information of the players, in second model we rejected the assumption that each player has only two strategies and reduced his choice to infinite (constant) strategic area, in third model we divided players into two subclasses using evolutional approach and we assumed short-sighted adjustment of both players, in fourth model, which is combination of second and third model, we assumed that one player is completely rational and the other one is short-sighted, fifth model reflects the rationality of both players but it rejects the perfect information assumption which means that player will know what are his payments for certain moves but he will not have completely cognition of other player payments. Mixed equilibrium each of these models leads to the same unique solution which we actually wanted to show in this master thesis, apropos that sanctions do not have influence and that they are not efficient on Nash equation properties in the game of the police and the criminals with assumptions from 1 to 4. Also the same theory was applied in Example 2, apropos in the sanction imposition game of one nation to another.