Praise be to Allah.
There is nothing wrong with making these games that are free of music and images of animate beings, for the purpose of advertising or other permissible purposes, provided that the promotion is based on facts and the description given of the product or medicine, and so on, is truthful.
There is nothing wrong with holding free contests (in which the competitor does not pay anything) to achieve the above, and to give the winner a stated prize. This comes under the heading of giving something in return for doing a task, which is permissible provided that what is to br given for is known. For example, it may be said: Whoever does the task in such and such a manner will be given such and such of money or goods and the like.
In al-Mawsoo‘ah al-Fiqhiyyah al-Kuwaitiyyah (15/216) it says: The Maalikis, Shaafa‘is and Hanbalis say: in order for a contract of giving something in return for a task to be valid, it is stipulated that what is to be given should be known in terms of type and amount, because not knowing what one will get defeats the purpose of the contract. Hardly anyone would want to work when he does not know what he will get. Moreover, there is no need to make it unknown in the contract. This is unlike the case of hiring a worker when there is ambiguity as to what the cost will be, because that is unavoidable. The information about what will be given when the task is completed may be attained by showing it or describing it, if it is an item, or describing it, if it is to be given later. End quote.
Undoubtedly this work may be useful in various fields, such as teaching, education, and da‘wah activities, where people may be told about some virtuous deed, or some of the rituals or symbols of Islam, of which many people are unaware, or some Islamic etiquette, or guiding people to do good deeds, or worldly benefits in general.
And Allah knows best.