Renewal of multiple notions of value in contemporary (green, public good) economy and are there general game theories?
The current mainstream theory of value is the subjective theory of value: goods or services have value that people subjectively believe that they have. Rembrandt paintings cost millions because someone subjectively thinks they are worth that much.

If there are no market failures the market price captures the subjective value through supply demand interactions.

However, in case of environment there are market failures. For example, due to lack of property rights to clean air firms use it for free and hence the subjective value of clean air is not reflected in market price. This would be example of negative externality. Externalities can be also positive, for example basic research can’t be patented so it’s value is more than scientist can get out of it by supplying their ideas on market.

If there is a negative or positive externality the subjective value is still there but it’s incredibly hard to find out. An example of how economists try to find this would be to use some econometric techniques like for example hedonic pricing which tries to estimate the value not captured in market price by observing how other market values change when availability of the good suffering the externality changes (for example by looking how pollution affect housing prices). Other techniques exists I would recommend starting with the hedonic pricing as it’s the most simple ones and you will see that many articles discussing it will point out to more complex techniques.

Also note that in many cases government might already correct for these externalities by taxes and subsidies or by issuing property rights over clean air by tradable polluting permits (cap and trade). Assuming, the government where you live already corrects for externalities and does it properly the market price would reflect also the subjective valuation of environmental goods etc.