Analysis of environmental art design practice and space utilization based on game theory perspective

Essentially, a coalition game involves a set of all participants, denoted by N$$\mathcal{N}$$, who seek to form cooperative groups to strengthen their position in a given situation. Any coalition S⊆N$$S \subseteq \mathcal{N}$$ of these represents participants in S interacting with each other as a separate entity. Another important concept is coalition utility, denoted as V$$\mathcal{V}$$, which is the same to measure the value of the coalition in the BoD. Nevertheless, independent of the definition of value, the only definition of a coalitional game is as follows:

Definition (1): a coalitional game (or coalitional form of the game) is defined by a pair of (N,V)$$(\mathcal{N},\mathcal{V})$$, where N$$\mathcal{N}$$ represents all the participants and V$$\mathcal{V}$$ represents a mapping of the utilities of the participants in the game.

In addition to this, there is another type of coalitional game known as transferable utility (TU). The definition of a Bo with the characteristic form of TU is as follows:

Definition (2): the characteristic function of a coalitional game with transferable utility is a function V$$\mathcal{V}$$ defined on the real numbers, denoted as: V:2N→ℝ$$\mathcal{V}:{2^\mathcal{N}} \to \mathbb{R}$$, V(∅)=0$$\mathcal{V}(\emptyset ) = 0$$.

This characteristic function is associated with each coalition S⊆N$$S \subseteq \mathcal{N}$$ and is a real number that quantifies the coalition’s benefits. The transferable utility property implies that the total utility represented by this real number can be distributed among the coalition members in any way. The members of a coalition can distribute it among themselves using appropriate fairness rules (one of which is the equal distribution of utility). The utility value that participant i ∈ S divides from V(S)$$\mathcal{V}(S)$$ constitutes that player’s payoff, denoted x_i. Vector X∈ℝS$$X \in {\mathbb{R}^S}$$ constitutes the set of utilities of the participants in coalition S, where each element x_i represents the utility value of participant i.

Although the TU characteristic function can model a considerable number of game types, in many cases the coalition values cannot be assigned to a single real number or there are strict restrictions on the distribution of utility. These games are referred to as coalitional games with non-transferable utility (NTU), where the payoff to each participant in coalition S depends on the joint decision chosen by the participants in coalition S. In NTU games, the utility V(S)$$\mathcal{V}(S)$$ of coalition S is no longer a function on real numbers, but a set of vectors V(S)∈ℝS$$\mathcal{V}(S) \in {\mathbb{R}^S}$$, where each element x_i of vector X∈V(S)$$X \in \mathcal{V}(S)$$ represents the payoffs that a participant i ∈ S can receive within the coalition under a certain strategy chosen while being a member of S. Based on this definition, TU bo’s can be considered as a particular case of the NTU framework. Coalitional games in the form of TU or NTU features constitute one of the most important types of cooperative games.

Normative Coalitional Gaming In as a type of coalitional game, coalitional utility can be either transferable or non-transferable, and its cooperation is always beneficial to all. That is, it is assumed that when forming a larger coalition, players do not obtain worse outcomes than if they act alone. For NTU, given any two disjoint coalitions S₁ and S₂, if the two coalitions can be combined, then the members within that larger coalition will always gain what they would have gained within each of the two coalitions had they not joined together into a larger coalition. This property can be corresponded to superadditivity in mathematics, defined as follows.

Definition (3): a union of NTUs with (N,V)$$(\mathcal{N},\mathcal{V})$$ is said to satisfy superadditivity if and only if the condition of Eq. (2) is satisfied. (2)V(S1∪S2)⊃{X∈ℝs1∪S2|(xi)i∈S1∈V(S1),(xj)j∈S2∈V(S2)}$$\mathcal{V}\left( {{S_1} \cup {S_2}} \right) \supset \left\{ {X \in {\mathbb{R}^{{s_1} \cup {S_2}}}|{{\left( {{x_i}} \right)}_{i \in {S_1}}} \in \mathcal{V}\left( {{S_1}} \right),{{\left( {{x_j}} \right)}_{j \in {S_2}}} \in \mathcal{V}\left( {{S_2}} \right)} \right\}$$

where ∀S1⊂N$$\forall {S_1} \subset \mathcal{N}$$, S2⊂N$${S_2} \subset \mathcal{N}$$, S₁ ∩ S₂ = ∅, and X are the utility allocation vectors of coalition S₁ ∪ S₂.

And for a TU game, the superadditivity given in Eq. (3) can be expressed as Eq. (3): (3)V(S1∪S2)≥V(S1)+V(S2),∀S1⊂N,S2⊂N,S1∩S2=∅$$\mathcal{V}\left( {{S_1} \cup {S_2}} \right) \geq \mathcal{V}\left( {{S_1}} \right) + \mathcal{V}\left( {{S_2}} \right),\forall {S_1} \subset \mathcal{N},{S_2} \subset \mathcal{N},{S_1} \cap {S_2} = \emptyset$$

The concept of the super-additivity game can be better understood from equation (3). The basic principle behind super-additivity is that in a coalition, participants can always fall back to their non-cooperative behavior to gain what they had when they made their non-cooperative decisions. Thus, in a super-additivity game, cooperation is never to the disadvantage of any of the participants.

The most well-known solution concept for the normative coalition game is the core. The core of a normative game is directly related to the stability of the grand coalition. In the normative coalition game (N,V)$$(\mathcal{N},\mathcal{V})$$, participants have an incentive to form a grand coalition due to the presence of superadditivity N$$\mathcal{N}$$. In this respect, the core of a typical game is a set of payoff allocations that ensures that no set of participants has an incentive to leave the coalition N$$\mathcal{N}$$ in order to form another coalition S⊆N$$S \subseteq \mathcal{N}$$. For the TU game, given grand coalition N$$\mathcal{N}$$, make the following definition.

Definition (4): For a gain division V(N)$$\mathcal{V}(\mathcal{N})$$, if ∑i∈Nxi=V(N)$$\sum\limits_{i \in \mathcal{N}} {{x_i}} = \mathcal{V}(\mathcal{N})$$, then the utility vector X∈ℝN$$X \in {\mathbb{R}^\mathcal{N}}$$ is said to satisfy collective rationality.

Definition (5): the utility vector X∈ℝN$$X \in {\mathbb{R}^\mathcal{N}}$$ is said to satisfy individual rationality if each participant receives no less than the gain of acting alone, i.e., xi≥V({i})$${x_i} \geq \mathcal{V}(\{ i\} )$$, ∀i∈N$$\forall i \in \mathcal{N}$$.

Based on the above definition of allocation, the core definition of the TU normative coalitional game can be given as shown below.

Definition (6): given a TU normative coalition game (N,V)$$(\mathcal{N},\mathcal{V})$$, its core definition is a set of sets of payoff allocations for which no coalition S⊆N$$S \subseteq \mathcal{N}$$ has an incentive to reject the offer. In mathematical form, the core CTU$${\mathcal{C}_{TU}}$$ of a TU normative coalition game can be expressed as equation (4): (4)CTU={X:∑i∈Nxi=V(N)and∑i∈Sxi≥V(S),∀S⊆N}$${\mathcal{C}_{TU}} = \left\{ {X:\sum\limits_{i \in \mathcal{N}} {{x_i}} = \mathcal{V}(\mathcal{N}){\text{ }}and{\text{ }}\sum\limits_{i \in S} {{x_i}} \geq \mathcal{V}(S),\forall S \subseteq \mathcal{N}} \right\}$$

In other words, the core guarantees that participants have no incentive to deviate from the grand coalition, since any distribution of gains X in the core guarantees utility at least equal to V(S)$$\mathcal{V}(S)$$ for every S⊆N$$S \subseteq \mathcal{N}$$. Clearly, as long as a participant can find a revenue distribution that is in the core, then the grand coalition is a stable and optimal solution to that coalitional game.

And for an NTU canonical coalition game (N,V)$$(\mathcal{N},\mathcal{V})$$, the core can be defined as equation (5): (5)CNTU={X∈V(N)|∀S,∄Y∈V(S),suchthatyi>xi,∀i∈S}$${\mathcal{C}_{NTU}} = \left\{ {X \in \mathcal{V}(\mathcal{N})|\forall S,\nexists Y \in \mathcal{V}(S),{\text{ }}such{\text{ }}that{\text{ }}{y_i} > {x_i},\forall i \in S} \right\}$$

This definition of NTU also guarantees a stable grand coalition. The basic idea is that in the core of the NTU game, any distribution of gains guarantees that no coalition S can leave the grand coalition with a better distribution for all its members. The difference with the case of the TU game is that in the core of the NTU game, the stability of the grand coalition is obtained based on the elements of the payoff vector, whereas in the TU game, the stability of the grand coalition is obtained through the sum of the elements of the payoff vector.

Nash equilibrium is widely used in game theory, this theory refers to a state of equilibrium that will be created in a cooperative game under uncertainty, the theory suggests that an equilibrium will be created when the type of game chosen by the participants forms a Nash equilibrium, in which case any player will change his or her game strategy to go for that Nash equilibrium, thus lowering his or her payoffs, and thus creating an equilibrium state.

A game with a number of participants, where the number of people is set to n and the individual strategy of a particular participant i is denoted as S, the set of strategies S_i can be expressed as equation (6): (6)G={S1,S2,⋯,Sn−1,Sn;u1,u2,⋯,un−1,un}$$G = \left\{ {{S_1},{S_2}, \cdots ,{S_{n - 1}},{S_n};{u_1},{u_2}, \cdots ,{u_{n - 1}},{u_n}} \right\}$$

where u_i denotes the value of the BoG’s payoff.

We assume that (Si*,S2*,⋯,Si*,Si+1*,⋯,Sn−1*,Sn*)$$\left( {S_i^*,S_2^*, \cdots ,S_i^*,S_{i + 1}^*, \cdots ,S_{n - 1}^*,S_n^*} \right)$$ is the set of Nash equilibrium strategies for game G, then there are (7)ui(S1*,S2*,⋯,Si*,Si+1,⋯,Sn−1*,Sn*)≥ui(S1*,S2*,⋯,Si,,i+1*,⋯,Sn−1,Sn*)$${u_i}\left( {S_1^*,S_2^*, \cdots ,S_i^*,{S_{i + 1}}, \cdots ,{S_{n - 1}}^*,{S_n}^*} \right) \geq {u_i}\left( {S_1^*,S_2^*, \cdots ,{S_i},{,_{i + 1}}^*, \cdots ,{S_{n - 1}},{S_n}^*} \right)$$

where ui(S1*,S2*,⋯,Si*,Si+1*,⋯,Sn−1*,Sn*)$${u_i}\left( {S_1^*,{S_2}^*, \cdots ,S_i^*,{S_{i + 1}}^*, \cdots ,{S_{n - 1}}^*,{S_n}^*} \right)$$ is the value of the gain under (S1*,S2*,⋯,Si*,Si+1,⋯,Sn−1,Sn*)$$\left( {S_1^*,S_2^*, \cdots ,S_i^*,{S_{i + 1}}, \cdots ,{S_{n - 1}},{S_n}^*} \right)$$.

The entrance of the public space of the neighborhood is the face of the urban residential public space, not only is the residential neighborhood and the city inside and outside the access road, but also to show the image of the residential neighborhood window. A good, clear, and bright entrance design can enhance the recognizability of the residential community and the accessibility of traffic. The use of thematic structures and combined with natural landscape or artificial landscape design, beautification of the district, beautifying the city street space, while enhancing the taste of the district.

The entrance design of the public space of the neighborhood tends to be the developer’s choice of strategy, creating a modern living atmosphere through the use of unique design styles with strong visual impact elements, catering to the psychology of the residents, expanding market competitiveness, attracting more residents, and obtaining higher economic benefits.

The courtyard space in the modern sense is very different from the courtyard space of traditional residential houses, which is no longer a space within a family but a public space belonging to an aggregate. However, the public qualities of the courtyard of the traditional folk house are still an effective way to learn. The courtyard space is an internal space relative to the city and the whole residential area; it is an external space relative to the residences within the group, and it is a transitional space between the residences and the external space. The courtyard space provides residents with a strong sense of belonging and domain, which is the basis for the harmony of neighborly relations. The function of the courtyard space is ambiguous and polysemous, it is a place for the elderly to chat and play chess, and also a place for children to play. It can also be a place for people to gather and have fun, or a place for temporary parking. Courtyard space is also an important landscape elements, people through the window, the most direct landscape is the courtyard landscape. The courtyard space has a relatively clear spatial definition, but the space is not closed, but an effective enclosure, the space is an extension of the building, or as a foreground, or as a backdrop, or as a visual focus, each such limited open space is a complete entity, but also an inseparable part of the adjacent spaces and structures.

The design adopts different forms of architectural space combinations to recreate traditional courtyard spaces. It not only enriches the spatial interface of the neighborhood, but also creates more diverse humanized spaces, meets the needs of various living scenarios, and meets the interests of the owners’ group. The design is based on the owner’s choice strategy, which focuses on effective space enclosure to meet the social and leisure needs of the owner group, enhancing their living experience and improving their quality of life.

When people stand in the horizontal direction to perceive the space, they usually experience changes in the depth of field of near view, center view, and far view in their vision. In the design of space form, the focus is to design a good center view, through the design of the center view to make the space form a rich hierarchy. The small sculpture in the center of the residential area forms a medium view in the vision, and the sculpture in the residential area should be connected with children’s activities, allowing children to climb, climb and slide on it. The water feature as centerpiece in the residential area is not only for visual enjoyment, but it should also be a small water source for children to get close to.

The design is still based on the owner’s choice of strategy, combined with the human body visual design of multi-level water features, providing adults with a beautiful view at the same time for children to provide a safer and more convenient place to play in the water. Through the rich layered design, we can meet the comfort and functionality needs of different groups of residents in a variety of needs.

The simulation results of the wind environment in the cell after optimization are shown in Fig. 2.

Figure 2.

Simulation results of wind environment optimization

As can be seen from the optimization simulation results, again the pedestrian height of 1.5m at the moment of 14:00pm is used as a reference standard. Objectively, the average wind speed in most areas of the study area is 1.87m/s in summer, and the average wind speed after optimization is 0.53m/s higher than that before optimization, which belongs to a more comfortable range, and the average wind speed in the center of the district is 2.97m/s before optimization, and 3.08m/s after optimization, and the local wind speed is increased by 0.11m/s. The difference between the optimized wind environment in poorer areas and the surrounding wind environment is reduced, and the local wind environment is more comfortable. The wind environment in the local area becomes more comfortable due to a reduction in the difference between the optimized area and the surrounding wind environment. Subjectively, the wind speed increases in summer, and the residents are more satisfied with the comfort of the outdoor space environment in the neighborhood.

Conclusion: The selection strategy of public space between developers and owners guided by the game mechanism can better maintain the summer wind environment inside the neighborhood and keep the ventilation in summer in the actual design.

The cell PMV simulation results after optimization are shown in Fig. 3.

Figure 3.

SIMULATION results of PMV optimization

As can be seen from the optimization simulation results, the same pedestrian height of 1.5 m at the moment of 14:00 p.m. is used as a reference standard. The average PMV value of the outdoor space in the study area in summer is 2.63, and the average PMV value after optimization is 2.40, with a decrease of 0.23. The average PMV value around the water body on the south side of the cell is 2.23 before optimization, and the average PMV value after optimization is 1.63, with a decrease of 0.60. It can be seen that the PMV value of the outdoor spatial environment of the cell in summer is reduced as a whole, and the comfort of the outdoor space environment increases.

Conclusion: the selection strategy of the developer and the owner of the public space under the guidance of the game mechanism can, to a certain extent, reduce the overall PMV value in summer in the actual design, and at the same time, the PMV of the local outdoor space is optimized, and the optimization effect in summer is more obvious.